0.08/0.15 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.08/0.15 % Command : tptp2X_and_run_prover9 %d %s 0.16/0.37 % Computer : n018.cluster.edu 0.16/0.37 % Model : x86_64 x86_64 0.16/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.16/0.37 % Memory : 8042.1875MB 0.16/0.37 % OS : Linux 3.10.0-693.el7.x86_64 0.16/0.37 % CPULimit : 1440 0.16/0.37 % WCLimit : 180 0.16/0.37 % DateTime : Mon Jul 3 10:16:46 EDT 2023 0.16/0.37 % CPUTime : 5.18/5.24 ============================== Prover9 =============================== 5.18/5.24 Prover9 (32) version 2009-11A, November 2009. 5.18/5.24 Process 18349 was started by sandbox2 on n018.cluster.edu, 5.18/5.24 Mon Jul 3 10:16:50 2023 5.18/5.24 The command was "/export/starexec/sandbox2/solver/bin/prover9 -t 1440 -f /tmp/Prover9_18194_n018.cluster.edu". 5.18/5.24 ============================== end of head =========================== 5.18/5.24 5.18/5.24 ============================== INPUT ================================= 5.18/5.24 5.18/5.24 % Reading from file /tmp/Prover9_18194_n018.cluster.edu 5.18/5.24 5.18/5.24 set(prolog_style_variables). 5.18/5.24 set(auto2). 5.18/5.24 % set(auto2) -> set(auto). 5.18/5.24 % set(auto) -> set(auto_inference). 5.18/5.24 % set(auto) -> set(auto_setup). 5.18/5.24 % set(auto_setup) -> set(predicate_elim). 5.18/5.24 % set(auto_setup) -> assign(eq_defs, unfold). 5.18/5.24 % set(auto) -> set(auto_limits). 5.18/5.24 % set(auto_limits) -> assign(max_weight, "100.000"). 5.18/5.24 % set(auto_limits) -> assign(sos_limit, 20000). 5.18/5.24 % set(auto) -> set(auto_denials). 5.18/5.24 % set(auto) -> set(auto_process). 5.18/5.24 % set(auto2) -> assign(new_constants, 1). 5.18/5.24 % set(auto2) -> assign(fold_denial_max, 3). 5.18/5.24 % set(auto2) -> assign(max_weight, "200.000"). 5.18/5.24 % set(auto2) -> assign(max_hours, 1). 5.18/5.24 % assign(max_hours, 1) -> assign(max_seconds, 3600). 5.18/5.24 % set(auto2) -> assign(max_seconds, 0). 5.18/5.24 % set(auto2) -> assign(max_minutes, 5). 5.18/5.24 % assign(max_minutes, 5) -> assign(max_seconds, 300). 5.18/5.24 % set(auto2) -> set(sort_initial_sos). 5.18/5.24 % set(auto2) -> assign(sos_limit, -1). 5.18/5.24 % set(auto2) -> assign(lrs_ticks, 3000). 5.18/5.24 % set(auto2) -> assign(max_megs, 400). 5.18/5.24 % set(auto2) -> assign(stats, some). 5.18/5.24 % set(auto2) -> clear(echo_input). 5.18/5.24 % set(auto2) -> set(quiet). 5.18/5.24 % set(auto2) -> clear(print_initial_clauses). 5.18/5.24 % set(auto2) -> clear(print_given). 5.18/5.24 assign(lrs_ticks,-1). 5.18/5.24 assign(sos_limit,10000). 5.18/5.24 assign(order,kbo). 5.18/5.24 set(lex_order_vars). 5.18/5.24 clear(print_given). 5.18/5.24 5.18/5.24 % formulas(sos). % not echoed (5482 formulas) 5.18/5.24 5.18/5.24 ============================== end of input ========================== 5.18/5.24 5.18/5.24 % From the command line: assign(max_seconds, 1440). 5.18/5.24 5.18/5.24 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 5.18/5.24 5.18/5.24 % Formulas that are not ordinary clauses: 5.18/5.24 1 (all X_2 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,X_2)),hAPP_int_real(real_int,Y_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Y_2)))) # label(fact_4387_real__of__int__less__iff) # label(axiom) # label(non_clause). [assumption]. 5.18/5.24 2 (all M_17 all N_30 ((zero_zero_nat != N_30 -> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,M_17),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,M_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_30),one_one_nat))) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,M_17),N_30)) & (N_30 = zero_zero_nat -> one_one_real = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,M_17),N_30)))) # label(fact_1831_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 5.18/5.24 3 (all D all Sa exists Z all X (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,Z),X)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,D),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X),Sa))) <-> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,D),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X),Sa)))))) # label(fact_2245_pinf_I9_J) # label(axiom) # label(non_clause). [assumption]. 5.18/5.24 4 (all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1)) -> zero_zero_real = hAPP_real_real(cos,arg(hAPP_real_complex(hAPP_r265291036omplex(complex,zero_zero_real),Y_1))))) # label(fact_3969_cos__arg__i__mult__zero__pos) # label(axiom) # label(non_clause). [assumption]. 5.18/5.24 5 (all Lx_3 all Ly_1 all Rx_3 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_3),Rx_3)),Ly_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_3),Ly_1)),Rx_3)) # label(fact_159_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 5.18/5.24 6 (all A_230 all M_29 all B_163 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_230),M_29)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_163),M_29)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_230),B_163)),M_29)) # label(fact_302_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.18/5.24 7 (all N_41 all A_190 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),A_190)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_190),N_41))))) # label(fact_861_one__le__power) # label(axiom) # label(non_clause). [assumption]. 5.18/5.24 8 (all X_1 all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_int_real(real_int,A_3))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim856651990g_real(X_1)),A_3)))) # label(fact_4422_ceiling__le) # label(axiom) # label(non_clause). [assumption]. 5.18/5.24 9 (all B_2 all Q_2 all R_3 all B_5 all Q_8 all R_5 (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_2)),R_3) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_5),Q_8)),R_5) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_5),Q_8)),R_5)),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),B_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_5),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_8),Q_2))))))))) # label(fact_1494_zdiv__mono2__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 5.18/5.24 10 (all N_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N_1)) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N_1),one_one_int)),hAPP_int_int(fact_fact_int,N_1)) = hAPP_int_int(fact_fact_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N_1),one_one_int)))) # label(fact_4286_fact__plus__one__int) # label(axiom) # label(non_clause). [assumption]. 5.18/5.24 11 (all A_228 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_228),one_one_rat) = A_228) # label(fact_333_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 5.18/5.24 12 (all M_14 all D_4 (hAPP_nat_nat(div_mod_nat(M_14),D_4) = zero_zero_nat -> (exists Q_4 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,D_4),Q_4) = M_14))) # label(fact_2780_mod__eq__0D) # label(axiom) # label(non_clause). [assumption]. 5.18/5.24 13 (all A_91 A_91 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_91),one_one_real)) # label(fact_2102_mult_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 5.18/5.24 14 (all Ma all X_2 (zero_zero_int = hAPP_int_int(standardRes(Ma),X_2) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_2),zero_zero_int),Ma)))) # label(fact_2446_StandardRes__eq__zcong) # label(axiom) # label(non_clause). [assumption]. 5.18/5.24 15 (all N_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(fact_fact_nat,N_1)))) # label(fact_3806_fact__gt__zero__nat) # label(axiom) # label(non_clause). [assumption]. 5.18/5.24 16 (all A_233 all C_93 all D_18 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_233),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_93),D_18)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_93),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_233),D_18))) # label(fact_238_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 17 (all B_2 all M all A_3 (is_int(B_2) & is_int(A_3) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M)) -> A_3 = B_2))))))) # label(fact_1486_zcong__zless__imp__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 18 (all X_2 (-hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven)) <-> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zOdd)))) # label(fact_2904_odd__iff__not__even) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 19 (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X_2)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),natfloor(X_2))))) # label(fact_3399_le__natfloor__eq__one) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 20 (all A_1 all P_2 all P_1 all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1)) -> ((exists Z all X (is_int(X) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z),X)) -> (hBOOL(hAPP_int_bool(P_1,X)) <-> hBOOL(hAPP_int_bool(P_2,X)))))) -> ((all X (is_int(X) -> ((all Xa (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) -> (all Xb (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A_1)) -> X != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa))))) -> (hBOOL(hAPP_int_bool(P_1,X)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),D_1))))))) -> ((all X all K (is_int(X) & is_int(K) -> (hBOOL(hAPP_int_bool(P_2,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D_1)))) <-> hBOOL(hAPP_int_bool(P_2,X))))) -> ((exists X (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) & (exists Xa (is_int(Xa) & hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xa),X))) & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),A_1)))) & is_int(X))) | (exists X (hBOOL(hAPP_int_bool(P_2,X)) & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) & is_int(X))) <-> (exists X1 (hBOOL(hAPP_int_bool(P_1,X1)) & is_int(X1))))))))) # label(fact_5155_cppi) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 21 (all X_1 all Y_1 all M (one_one_int = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_1)),M) -> one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),M))) # label(fact_4962_invertible__coprime__int) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 22 (all A all B all K_1 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ma)) -> (one_one_int = hAPP_int_int(legacy_zgcd(K_1),Ma) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),A)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),B)),Ma)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma)))))) # label(fact_4525_zcong__cancel2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 23 (all A all E_2 all C all B all D (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),E_2)),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),E_2)),D))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B),A)),E_2)),D))))) # label(fact_1576_less__add__iff2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 24 (all C_37 all A_94 all B_77 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_94),B_77)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_94),C_37)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_77),C_37))))) # label(fact_2061_add__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 25 (all C all A all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C),A)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C),B))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),B)))) # label(fact_2068_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 26 (all Ma all N all A (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A)) -> (hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A),Ma) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A),N) <-> N = Ma))) # label(fact_874_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 27 (all Z_4 all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),zero_zero_int)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(nat,hAPP_int_int(uminus_uminus_int,Z_1))),hAPP_int_nat(nat,hAPP_int_int(uminus_uminus_int,Z_4))) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),Z_4)))) # label(fact_3550_nat__mult__distrib__neg) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 28 (all A_225 all M_28 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_225),M_28)),M_28) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_225),one_one_int)),M_28)) # label(fact_392_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 29 (all X_1 all Y_1 (Y_1 != X_1 -> (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_1),X_1))))) # label(fact_1685_linorder__neqE__nat) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 30 (all Z_1 all W hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_1),W) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,W),Z_1)) # label(fact_1596_real__mult__commute) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 31 (all X_2 all P_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(standardRes(P_5),X_2)),hAPP_i1948725293t_bool(sRStar,P_5))) <-> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_2),zero_zero_int),P_5))))) # label(fact_2527_StandardRes__SRStar__prop1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 32 (all A_106 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_106),zero_zero_rat) = A_106) # label(fact_1976_add__0__right) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 33 (all X_1 all Y_1 hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,X_1))),hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,Y_1)))) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_1)) # label(fact_5020_gcd__int__def) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 34 (all X_1 all Y_1 hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),hAPP_int_int(uminus_uminus_int,Y_1)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_1)) # label(fact_4993_gcd__neg2__int) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 35 (all F all F_1 all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)) -> ((all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),B)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X)) -> hBOOL(hAPP_real_bool(deriv_real(F,X),hAPP_real_real(F_1,X))))) -> (exists Z (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),Z)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z),B)) & hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B),A)),hAPP_real_real(F_1,Z)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,B)),hAPP_real_real(F,A))))))) # label(fact_3944_MVT2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 36 (all X_1 all A_3 hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3)) = hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),hAPP_real_real(uminus_uminus_real,A_3))) # label(fact_4413_powr__minus) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 37 (all V_20 all V_19 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_19)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_20)) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_19)),hAPP_int_int(number_number_of_int,V_20)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_19),V_20))))) # label(fact_372_semiring__add__number__of) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 38 (all P_1 all I_1 all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),K_1)) -> (hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_1),one_one_int))) -> ((all I_2 (is_int(I_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_2),K_1)) -> (hBOOL(hAPP_int_bool(P_1,I_2)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,I_2),one_one_int))))))) -> hBOOL(hAPP_int_bool(P_1,I_1)))))) # label(fact_2233_int__less__induct) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 39 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,pi)),X_1)) -> hAPP_real_real(arccos,hAPP_real_real(cos,X_1)) = hAPP_real_real(uminus_uminus_real,X_1)))) # label(fact_3608_arccos__cos2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 40 (all V_2 all W ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W))) -> hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,V_2))),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_2)),one_one_int)),hAPP_int_int(number_number_of_int,W)))),one_one_int)) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W))) -> hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,V_2))),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,W)))),one_one_int)))) # label(fact_2293_zmod__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 41 (all P_1 all X_2 all Y_2 (hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,X_2),Y_2)))) <-> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),Y_2)) -> hBOOL(hAPP_int_bool(P_1,zero_zero_int))) & (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_2),X_2)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(semiri1621563631at_int,X_2)),hAPP_nat_int(semiri1621563631at_int,Y_2))))))) # label(fact_3195_zdiff__int__split) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 42 (all P_3 all X_1 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_int_int(multInv(P_3),X_1)) # label(fact_2770_MultInv__def) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 43 (all N_66 all A_217 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_217)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_217),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_217),N_66)))))) # label(fact_522_power__gt1__lemma) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 44 (all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(zfact,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))),hAPP_int_int(number_number_of_int,min)),P_3)))) # label(fact_1855_Wilson__Russ) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 45 (all X_20 all Y_16 all A_77 all B_66 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_20),Y_16)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_77),B_66)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_20),A_77)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_16),B_66))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_20),A_77)),B_66))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_77),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_16),B_66)))) # label(fact_2289_mult_Oprod__diff__prod) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 46 (all V_4 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_4))) -> hAPP_int_int(number_number_of_int,V_4) = hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(number_number_of_nat,V_4))) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_4))) -> zero_zero_int = hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(number_number_of_nat,V_4))))) # label(fact_3221_of__nat__number__of__lemma) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 47 (all V_9 all U_2 all Y_20 all X_26 all A_122 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_26),A_122)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_20),A_122)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),U_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),V_9)) -> (one_one_real = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,U_2),V_9) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,U_2),X_26)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,V_9),Y_20))),A_122)))))))) # label(fact_1491_convex__bound__le) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 48 (all F all A_1 hBOOL(hAPP_f448129468l_bool(nat_tr876908586nt_nat(F),A_1))) # label(fact_2787_transfer__morphism__def) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 49 (all Z_1 Z_1 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,one_one_real),Z_1)) # label(fact_1595_real__mult__1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 50 (all X_46 X_46 = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_46),one_one_nat)) # label(fact_80_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 51 (all Ma all N (zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),N) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)))) # label(fact_1661_diff__is__0__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 52 (all A_108 all B_87 all C_47 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_108),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_87),C_47)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_108),B_87)),C_47)) # label(fact_1947_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 53 (all X_31 all Y_24 (Y_24 != X_31 -> (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_31),Y_24)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_24),X_31))))) # label(fact_970_linorder__neqE__linordered__idom) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 54 (all B_125 all A_149 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_149)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_125)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_149),B_125)))))) # label(fact_1243_mult__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 55 (all V_5 all W_3 all C_54 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_5)),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_int_complex(number528085621omplex,W_3)),C_54)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_5),W_3))),C_54)) # label(fact_1583_add__number__of__diff1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 56 (all P_5 all A (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),hAPP_i1948725293t_bool(wset(A),P_5))))) # label(fact_2236_wset__mem__mem) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 57 (all Z_1 (is_int(Z_1) -> hAPP_int_int(uminus_uminus_int,hAPP_int_int(uminus_uminus_int,Z_1)) = Z_1)) # label(fact_3517_zminus__zminus) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 58 (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_2)) -> hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_r474017924t_real(power_power_real,X_2)),zero_zero_real,sequentially)))) # label(fact_5044_LIMSEQ__inverse__realpow__zero) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 59 (all B_74 all A_85 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_85),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_74),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_85),B_74)),zero_zero_int))))) # label(fact_2149_add__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 60 (all Q_2 all B_2 all R_3 all C_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(div_mod_int(Q_2),C_1))),R_3))))))) # label(fact_2463_zmult2__lemma__aux3) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 61 (all X_2 all Y_2 (X_2 = Y_2 <-> hAPP_complex_complex(cnj,Y_2) = hAPP_complex_complex(cnj,X_2))) # label(fact_4070_complex__cnj__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 62 (all B_127 all A_151 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_151)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_127),zero_zero_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_127),A_151)),zero_zero_rat))))) # label(fact_1232_mult__nonneg__nonpos2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 63 (all X_1 all Y_1 hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,ratreal(X_1)),ratreal(Y_1)) = ratreal(hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,X_1),Y_1))) # label(fact_4703_real__divide__code) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 64 (all A_3 all B_2 all M hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),B_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),M)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),M)))),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(div_mod_int(A_3),M)),hAPP_int_int(div_mod_int(B_2),M)))) # label(fact_2691_zcong__zmod__aux) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 65 (all A_209 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_209),zero_z126310315umeral) = zero_z126310315umeral) # label(fact_618_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 66 (all P_1 all I_1 hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),P_1)),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),I_1)))))) # label(fact_4765_finite__M__bounded__by__nat) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 67 (all N_59 all A_212 (zero_zero_rat != A_212 -> zero_zero_rat != hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_212),N_59))) # label(fact_593_field__power__not__zero) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 68 (all N_12 all R_6 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),R_6)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,R_6),one_on1684967323de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,R_6),hAPP_nat_nat(suc,N_12))),R_6))))) # label(fact_3157_realpow__Suc__le__self) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 69 (all X_2 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X_2)),zero_zero_real))))) # label(fact_3916_real__root__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 70 (all X_2 all Y_2 (hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_2),Y_2))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Y_2)) & hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2)))) # label(fact_3776_even__power__nat) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 71 (all X_2 all Ma (hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_2),Ma) = hAPP_nat_nat(suc,zero_zero_nat) <-> X_2 = hAPP_nat_nat(suc,zero_zero_nat) | Ma = zero_zero_nat)) # label(fact_3086_nat__power__eq__Suc__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 72 (all X_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X_1)),hAPP_real_real(exp_real,X_1)))) # label(fact_3844_exp__ge__add__one__self) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 73 (all R_3 all X_1 hAPP_real_real(scaleR_scaleR_real(R_3),hAPP_complex_real(re,X_1)) = hAPP_complex_real(re,hAPP_complex_complex(scaleR1652505878omplex(R_3),X_1))) # label(fact_5110_Re_OscaleR) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 74 (all D all Sa exists Z all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Sa))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Sa)))))) # label(fact_2246_pinf_I9_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 75 (all M all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,M),N_1))) # label(fact_5123_min__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 76 (all A_172 all B_147 all C_85 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_172),B_147)),C_85)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_147),C_85)))) # label(fact_1091_dvd__mult__right) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 77 (all P all Q all R hAPP_f1375227553nt_int(P,hAPP_i1584592887nt_int(Q,R)) = hAPP_i1153444985nt_int(hAPP_f55007289nt_int(hAPP_f747475781nt_int(cOMBB_222539774nt_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__prod_Itc__Int__Oint_Mtc__Int_038) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 78 (all A_3 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(abs_abs_int,A_3)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(abs_abs_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3458_abs__power2__distrib) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 79 (all N_33 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_33)) = one_one_int) # label(fact_1169_power__m1__even) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 80 (all X_39 all Y_31 -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_39),X_39)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_31),Y_31))),zero_zero_real))) # label(fact_736_not__sum__squares__lt__zero) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 81 (all B_143 all A_168 all C_82 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_168),C_82)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_168),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_143),C_82))))) # label(fact_1113_dvd__mult) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 82 (all Y_1 all X_1 all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_3)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,Y_1),A_3))))))) # label(fact_4429_powr__mono2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 83 (all X_1 (hAPP_real_real(cos,X_1) = one_one_real -> hAPP_real_real(sin,X_1) = zero_zero_real)) # label(fact_3661_cos__one__sin__zero) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 84 (all C_59 all A_125 all B_104 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_59),A_125)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_59),B_104))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_59)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_125),B_104))))) # label(fact_1452_mult__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 85 (all K_1 all L (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K_1)),hAPP_int_int(bit1,L))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),L)))) # label(fact_130_rel__simps_I15_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 86 (all X_1 all Y_1 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_1)))) # label(fact_5008_gcd__ge__0__int) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 87 (all C_76 all B_133 all A_157 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_133),A_157)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_76),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_157),C_76)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_133),C_76)))))) # label(fact_1198_mult__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 88 (all K_1 all L hAPP_int_int(legacy_zgcd(K_1),L) = hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,L),zero_zero_int)),K_1),hAPP_int_int(legacy_zgcd(L),hAPP_int_int(div_mod_int(hAPP_int_int(abs_abs_int,K_1)),hAPP_int_int(abs_abs_int,L)))))) # label(fact_4519_zgcd__code) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 89 (all X_3 ((all R_1 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),R_1)) -> (exists K all N_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N_2)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,hAPP_nat_rat(X_3,N_2))),R_1)))))) <-> hBOOL(vanishes(X_3)))) # label(fact_5149_vanishes__def) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 90 (all X_1 all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),Y_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(tan,Y_1)),hAPP_real_real(tan,X_1))))))) # label(fact_3616_tan__monotone) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 91 (all C_34 all D_6 all A_79 all B_68 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_79),B_68)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_34),D_6)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_79),C_34)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_68),D_6)))))) # label(fact_2203_add__le__less__mono) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 92 (all A_36 hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_36),zero_zero_int) = zero_zero_int) # label(fact_2537_div__by__0) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 93 (all A_121 all C_56 all B_101 all D_9 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_121),B_101)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C_56),D_9)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_121),C_56)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_101),D_9))) # label(fact_1524_add__diff__add) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 94 (all P_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int) = hAPP_nat_int(semiri1621563631at_int,hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(sRStar,P_5))))) # label(fact_3266_SRStar__card) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 95 (all X_1 (zero_zero_real = hAPP_real_real(sin,X_1) -> hAPP_real_real(abs_abs_real,hAPP_real_real(cos,X_1)) = one_one_real)) # label(fact_3680_sin__zero__abs__cos__one) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 96 (all A hBOOL(tendsto_complex_real(re,hAPP_complex_real(re,A),at_complex(A)))) # label(fact_5075_Re_Ocont) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 97 (all X_2 ((exists N_2 (hAPP_int_real(real_int,N_2) = X_2 & is_int(N_2))) <-> hAPP_int_real(real_int,archim856651990g_real(X_2)) = X_2)) # label(fact_4397_real__of__int__ceiling__cancel) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 98 (all V_3 ((-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_3))) -> hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,zero_zero_nat)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_int_nat(nat,hAPP_int_int(div_mod_int(one_one_int),hAPP_int_int(number_number_of_int,V_3)))) & (hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_3))) -> hAPP_nat_nat(suc,zero_zero_nat) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,zero_zero_nat)),hAPP_int_nat(number_number_of_nat,V_3))))) # label(fact_3265_one__mod__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 99 (all C_64 all D_11 all A_130 all B_109 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_130),B_109)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,C_64),D_11)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_130)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_64)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_130),C_64)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_109),D_11)))))))) # label(fact_1422_mult__less__le__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 100 (all A_39 all B_32 all B_31 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_39),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,B_32),B_31)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_39),B_32)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_39),B_31))) # label(fact_2487_mult_Odiff__right) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 101 (all D all F all X_2 all L (hBOOL(hAPP_real_bool(deriv_real(F,X_2),L)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D)) -> ((all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_2),Y))),D)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(F,X_2)),hAPP_real_real(F,Y))))) -> L = zero_zero_real)))) # label(fact_3884_DERIV__local__min) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 102 (all N_1 all K_2 all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),M)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)))))) # label(fact_1643_dvd__diff__nat) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 103 (exists K_4 all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(cnj,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X)),K_4)))) # label(fact_4087_cnj_Obounded) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 104 (all M all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),M))))) # label(fact_4160_prime__dvd__power__two) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 105 (all B_72 all C_35 all A_83 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_83)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_72),C_35)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_72),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_83),C_35)))))) # label(fact_2164_add__increasing) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 106 (all R_3 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,archim1246769320r_real(R_3))),one_one_real)))) # label(fact_4452_real__of__int__floor__add__one__gt) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 107 (all V_2 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),one_one_nat) = one_one_nat) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls)) -> hAPP_int_nat(number_number_of_nat,hAPP_int_int(succ,V_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),one_one_nat)))) # label(fact_3303_nat__number__of__add__1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 108 (all B_128 all A_152 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_152),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_128)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_152),B_128)),zero_zero_real))))) # label(fact_1223_mult__nonpos__nonneg) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 109 (all A_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),one_one_nat)) -> one_one_nat = A_3)) # label(fact_2279_gcd__lcm__complete__lattice__nat_Ole__bot) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 110 (all M_17 all N_30 ((N_30 = zero_zero_nat -> one_one_rat = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,M_17),N_30)) & (zero_zero_nat != N_30 -> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,M_17),N_30) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,M_17),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,M_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_30),one_one_nat)))))) # label(fact_1835_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 111 (all M_19 all N_38 all A_188 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_188)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_188),M_19)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_188),N_38))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_19),N_38))))) # label(fact_888_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 112 (all X_1 one_one_real = hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),zero_zero_real)) # label(fact_4407_powr__zero__eq__one) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 113 (all W hAPP_int_int(ring_1_of_int_int,W) = hAPP_int_int(number_number_of_int,W)) # label(fact_4581_int__number__of__def) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 114 (all A_173 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,zero_z891286103de_int),A_173)) -> A_173 = zero_z891286103de_int)) # label(fact_1081_dvd__0__left) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 115 (all C_74 all A_155 all B_131 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_155),B_131)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_74)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_74),A_155)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_74),B_131)))))) # label(fact_1208_mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.34/5.38 116 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)))) # label(fact_2866_Suc__lessD) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 117 (all X_1 all Y_1 all C_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_1),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_1),hAPP_real_real(abs_abs_real,Y_1)))) -> zero_zero_real = X_1))) # label(fact_3393_rabs__ratiotest__lemma) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 118 (all X_21 all N_31 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31)) -> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_21),N_31) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_21),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_31),one_one_nat))),X_21))) # label(fact_1737_realpow__minus__mult) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 119 (all A_198 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_198),one_one_nat) = A_198) # label(fact_750_power__one__right) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 120 (all X_1 all Y_1 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_1)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_1),Y_1))) # label(fact_3041_Nat__Transfer_Otransfer__int__nat__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 121 (all X_2 (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_2),X_2))) <-> X_2 = zero_zero_real)) # label(fact_933_not__real__square__gt__zero) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 122 (all P_5 (is_int(P_5) -> (hBOOL(hAPP_int_bool(zprime,P_5)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),P_5)) & (all M_1 (is_int(M_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M_1),P_5)) -> P_5 = M_1 | M_1 = one_one_int)))))) # label(fact_902_zprime__def) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 123 (all M all N_1 all Q_2 all R_3 (hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Q_2),R_3))) -> hAPP_nat_nat(div_mod_nat(M),N_1) = R_3)) # label(fact_4218_mod__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 124 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(cos,X_1)))))) # label(fact_3590_cos__gt__zero__pi) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 125 (all C_29 all A_69 all B_61 hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_29),A_69)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_29),B_61)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_29),hAPP_Q1762011733de_int(div_mo231679042de_int(A_69),B_61))) # label(fact_2329_mod__mult__mult1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 126 (all A_225 all M_28 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_225),M_28)),M_28) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_225),one_one_nat)),M_28)) # label(fact_393_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 127 (all Z_2 all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,Z_2))),Ma)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_2),hAPP_nat_int(semiri1621563631at_int,Ma))))) # label(fact_3449_dvd__int__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 128 (all B_90 all A_111 all C_50 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_90),A_111) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_50),A_111) -> C_50 = B_90)) # label(fact_1912_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 129 (all C all D all A all B (hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),D)) != hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),D)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),C)) <-> C != D & B != A)) # label(fact_319_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 130 (exists K_4 all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(re,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X)),K_4)))) # label(fact_4141_Re_Obounded) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 131 (all A (A != zero_zero_nat <-> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),zero_zero_nat)))) # label(fact_2276_gcd__lcm__complete__lattice__nat_Oless__top) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 132 (all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),zero_zero_real)) -> zero_zero_real = hAPP_real_real(cos,arg(hAPP_real_complex(hAPP_r265291036omplex(complex,zero_zero_real),Y_1))))) # label(fact_3968_cos__arg__i__mult__zero__neg) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 133 (all Ma all Z_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(semiri1621563631at_int,Ma)),Z_2)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,Z_2)))))) # label(fact_3450_int__dvd__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 134 (all P_1 all N all K_1 ((zero_zero_nat = K_1 -> hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))) & (zero_zero_nat != K_1 -> (all I_2 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),K_1)) -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),I_2)),J_1) = N -> hBOOL(hAPP_nat_bool(P_1,I_2)))))) <-> hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N),K_1))))) # label(fact_2748_split__div) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 135 (all C all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C),A)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C),B))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)))) # label(fact_2027_add__le__cancel__left) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 136 (all X_1 all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(standardRes(P_3),X_1))))) # label(fact_2459_StandardRes__lbound) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 137 (all M_17 all N_30 ((N_30 != zero_zero_nat -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,M_17),N_30) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M_17),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,M_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_30),one_one_nat)))) & (N_30 = zero_zero_nat -> one_one_int = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,M_17),N_30)))) # label(fact_1829_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 138 (all N_35 all A_182 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_182)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_35)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_182),N_35)))))) # label(fact_972_one__less__power) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 139 (all M all K_2 all N_1 (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,K_2),N_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N_1))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M))))) # label(fact_4965_coprime__dvd__mult__int) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 140 (all X_31 all Y_24 (is_int(Y_24) & is_int(X_31) -> (Y_24 != X_31 -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_31),Y_24)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_24),X_31)))))) # label(fact_969_linorder__neqE__linordered__idom) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 141 (all A_73 zero_z891286103de_int = hAPP_Q1762011733de_int(div_mo231679042de_int(zero_z891286103de_int),A_73)) # label(fact_2313_mod__0) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 142 (all A_3 all B_2 all C_1 hAPP_nat_nat(div_mod_nat(A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),C_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_3),B_2)),C_1))),hAPP_nat_nat(div_mod_nat(A_3),B_2))) # label(fact_2744_mod__mult2__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 143 (all A all B all C (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B),A)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),A)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B)))))) # label(fact_1288_mult__less__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 144 (all X_2 all A (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,A)),one_one_real))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim1246769320r_real(X_2)),A)))) # label(fact_4468_floor__le__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 145 (all V all W_1 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(number_number_of_rat,V)),hAPP_int_rat(number_number_of_rat,W_1))) <-> -hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(number_number_of_rat,W_1)),hAPP_int_rat(number_number_of_rat,V))))) # label(fact_89_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 146 (all Z_12 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_12) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_12),Z_12)) # label(fact_487_semiring__mult__2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 147 (all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),N_1) = zero_zero_nat) # label(fact_1640_diff__0__eq__0) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 148 (all C all B all A (zero_zero_nat != A -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B)) -> (C = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B),A) <-> B = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C),A))))) # label(fact_2666_dvd__div__eq__mult) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 149 (all E_1 all B_2 all D_3 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_3),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,E_1),B_2)) -> (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2) -> one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),E_1))))) # label(fact_4974_coprime__divisors__nat) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 150 (all A_173 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A_173)) -> zero_zero_nat = A_173)) # label(fact_1084_dvd__0__left) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 151 (all X_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(exp_real,X_1)))) # label(fact_3834_exp__gt__zero) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 152 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) -> divmod_nat(M,N_1) = hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,zero_zero_nat),M))) # label(fact_4209_divmod__nat__base) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 153 (all A_92 hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_92),A_92) = zero_zero_rat) # label(fact_2085_diff__self) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 154 (all P all Q all R hAPP_f883232462nt_int(hAPP_i998473677nt_int(P,R),Q) = hAPP_i1584592887nt_int(hAPP_f1574055885nt_int(hAPP_f960630521nt_int(cOMBC_1805044090nt_int,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_019) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 155 (all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),Y_2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,Y_2))))) # label(fact_762_le__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 156 (all K_2 all L_2 hAPP_int_int(bit0,L_2) != hAPP_int_int(bit1,K_2)) # label(fact_350_rel__simps_I50_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 157 (all Lx_5 all Ly_3 all Rx_5 all Ry_3 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_5),Ly_3)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Rx_5),Ry_3)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Rx_5),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_5),Ly_3)),Ry_3))) # label(fact_143_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 158 (all A_127 all C_61 all B_106 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_127),C_61)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_106),C_61))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_61)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_127),B_106))))) # label(fact_1441_mult__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 159 (all B_70 all A_81 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_81),zero_z126310315umeral)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_70),zero_z126310315umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_81),B_70)),zero_z126310315umeral))))) # label(fact_2184_add__neg__neg) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 160 (all N (hAPP_nat_int(semiri1621563631at_int,N) = zero_zero_int <-> N = zero_zero_nat)) # label(fact_2977_int__eq__0__conv) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 161 (all A (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) <-> zero_zero_rat != A)) # label(fact_800_zero__less__power2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 162 (all X_2 (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),field_1210416355s_real)) -> -(all M_1 all N_2 (N_2 != zero_zero_nat -> (hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,M_1)),hAPP_nat_real(real_nat,N_2)) = hAPP_real_real(abs_abs_real,X_2) -> one_one_nat != hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M_1),N_2)))))) # label(fact_4629_Rats__abs__nat__div__natE) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 163 (all I_1 (is_int(I_1) -> (zero_zero_int != I_1 -> hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,dvd_dvd_int),I_1))))))) # label(fact_4752_finite__divisors__int) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 164 (all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K_1)),min)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),min)))) # label(fact_966_rel__simps_I11_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 165 (all N_1 (-hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)))) # label(fact_3762_odd__pos) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 166 (all N_57 all A_211 all B_157 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_211),B_157)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_211)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_57)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_211),N_57)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_157),N_57))))))) # label(fact_607_power__strict__mono) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 167 (all A all C all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B),C))))) # label(fact_2075_add__less__cancel__right) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 168 (all A_3 all B_2 all C_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_3),B_2)),C_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_3),C_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B_2),C_1))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(div_mod_nat(A_3),C_1)),hAPP_nat_nat(div_mod_nat(B_2),C_1))),C_1))) # label(fact_2732_div__add1__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 169 (all N all K_1 all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),Ma)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),K_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)))))) # label(fact_1665_less__diff__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 170 (all Y_1 all X_1 (hBOOL(hAPP_int_bool(nat_is_nat,X_1)) -> (hBOOL(hAPP_int_bool(nat_is_nat,Y_1)) -> hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(nat_tsub(X_1),Y_1)))))) # label(fact_5137_Nat__Transfer_Otransfer__int__nat__function__closures_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 171 (all Y_1 (Y_1 != zero_zero_nat -> -(all Nat_1 hAPP_nat_nat(suc,Nat_1) != Y_1))) # label(fact_3353_nat_Oexhaust) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 172 (all M_9 all N_23 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(semiri1424489471de_int,M_9)),hAPP_n522471361de_int(semiri1424489471de_int,N_23))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_23)))) # label(fact_2945_of__nat__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 173 (all P all Q all R hAPP_real_bool(P,hAPP_real_real(Q,R)) = hAPP_real_bool(hAPP_f1402056515l_bool(hAPP_f824790735l_bool(cOMBB_real_bool_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__HOL__Obool_000tc__RealDef__Ore) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 174 (all C_44 all D_8 all A_101 all B_84 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_101),B_84)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,C_44),D_8)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_101),C_44)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_84),D_8)))))) # label(fact_2006_add__mono) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 175 (all N_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),hAPP_nat_nat(fact,N_1)))) # label(fact_4566_fact__le) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 176 (all V all K_1 all A all B (B != A -> ((all X hBOOL(hAPP_real_bool(deriv_real(V,X),K_1))) -> hAPP_real_real(V,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),B)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(V,A)),hAPP_real_real(V,B))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_3882_DERIV__const__average) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 177 (all N_1 all M hAPP_C2132160860umeral(hAPP_C1614127039umeral(produc2136830103umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,N_1),M)),hAPP_C498520661umeral(div_mo1740067990umeral(N_1),M)) = code_d418564891umeral(N_1,M)) # label(fact_4197_div__mod__code__numeral__def) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 178 (all A_3 all B_2 all C_1 all Q_2 all R_3 (is_int(C_1) -> (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(B_2,C_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3))) -> (C_1 != zero_zero_int -> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2),C_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),Q_2)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),R_3)),C_1))),hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),R_3)),C_1)))))))) # label(fact_2804_zmult1__lemma) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 179 (all N all Ma (is_int(N) & is_int(Ma) -> (hAPP_i1732201573de_int(quickcheck_of_int,Ma) = hAPP_i1732201573de_int(quickcheck_of_int,N) <-> Ma = N))) # label(fact_4553_Quickcheck__Narrowing_Oof__int__inject) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 180 (all K_2 hAPP_int_int(number_number_of_int,K_2) = quickcheck_int_of(hAPP_i1732201573de_int(number1226105091de_int,K_2))) # label(fact_4255_int__of__number) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 181 (all M all K_2 all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_2)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_2))),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,J_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_2),M)))) # label(fact_3140_diff__Suc__diff__eq2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 182 (all V_10 all U_3 all Y_25 all X_33 all A_185 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_33),A_185)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_25),A_185)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),U_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),V_10)) -> (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,U_3),V_10) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,U_3),X_33)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_10),Y_25))),A_185)))))))) # label(fact_906_convex__bound__lt) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 183 (all V all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))),hAPP_nat_nat(suc,N)),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))),N)))) # label(fact_5195_max__Suc__number__of) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 184 (all A_234 all C_94 all D_19 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_234),C_94)),D_19) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_234),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_94),D_19))) # label(fact_230_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 185 (all N_1 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N_1)),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N_1)),one_one_real))) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N_1),one_one_int) = archim856651990g_real(X_1)))) # label(fact_4375_ceiling__eq2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 186 (all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),N_1) = N_1) # label(fact_1748_plus__nat_Oadd__0) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 187 (all C_39 all D_7 all A_96 all B_79 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_96),B_79)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_39),D_7)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_96),C_39)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_79),D_7)))))) # label(fact_2053_add__strict__mono) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 188 (all L_2 all K_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_2),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),L_2)) -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,K_2),L_2)),hAPP_int_int(div_mod_int(K_2),L_2)) = negDivAlg(K_2,L_2)))) # label(fact_2698_negDivAlg__div__mod) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 189 (all K_1 all L (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_1),L)),zero_zero_int)))) # label(fact_1566_less__bin__lemma) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 190 (all Ma all N image_int_nat(nat,hAPP_i1948725293t_bool(ord_at875362053st_int(hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,N))) = hAPP_n1699378549t_bool(ord_at238088361st_nat(Ma),N)) # label(fact_5070_SetInterval_Otransfer__nat__int__set__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 191 (all V_4 ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_4))) -> hAPP_nat_complex(semiri2020571505omplex,hAPP_int_nat(number_number_of_nat,V_4)) = zero_zero_complex) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_4))) -> hAPP_int_complex(number528085621omplex,V_4) = hAPP_nat_complex(semiri2020571505omplex,hAPP_int_nat(number_number_of_nat,V_4))))) # label(fact_3219_of__nat__number__of__lemma) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 192 (all X_1 all Y_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Y_1)),Y_1))) # label(fact_4329_gcd__semilattice__nat_Oinf__le2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 193 (all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1)) -> hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),Z_1)) = hAPP_nat_nat(suc,hAPP_int_nat(nat,Z_1)))) # label(fact_3205_Suc__nat__eq__nat__zadd1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 194 (all N_1 all K_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),N_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N_1)))))) # label(fact_4984_gcd__greatest__int) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 195 (all A_174 all B_148 all C_86 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_174),C_86)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_148),C_86)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_174),B_148)),C_86)) # label(fact_1077_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 196 (all A_194 all B_153 all N_46 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_194),B_153)),N_46) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_194),N_46)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_153),N_46))) # label(fact_815_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 197 (all N_1 all M hAPP_i1732201573de_int(quickcheck_of_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,quickcheck_int_of(N_1)),quickcheck_int_of(M))) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(minus_534354567de_int,N_1),M)) # label(fact_4561_minus__code__int__def) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 198 (all Code_numeral_5 all Code_numeral_4 (code_S1047413653umeral(Code_numeral_4) = code_S1047413653umeral(Code_numeral_5) <-> Code_numeral_5 = Code_numeral_4)) # label(fact_4072_code__numeral_Oinject) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 199 (all N_1 all M hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_1)))),hAPP_nat_int(semiri1621563631at_int,M)))) # label(fact_3510_negative__zless) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 200 (all K_1 all Ma all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N))) <-> zero_zero_nat = K_1 | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),N)))) # label(fact_1846_nat__mult__dvd__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 201 (all A_230 all M_29 all B_163 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_230),M_29)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_163),M_29)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_230),B_163)),M_29)) # label(fact_304_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 202 (all A_3 all B_2 hAPP_int_int(div_mod_int(A_3),B_2) = hAPP_P1175774780nt_int(product_snd_int_int,divmod_int(A_3,B_2))) # label(fact_4232_mod__int__def) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 203 (all C_43 all A_100 all B_83 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_100),B_83)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_43),A_100)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_43),B_83))))) # label(fact_2017_add__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 204 (all M all N_1 all K_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)),K_2)) # label(fact_1657_diff__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 205 (all K_2 all M all N_1 (is_int(K_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N_1))) -> (K_2 != zero_zero_int -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),N_1)))))) # label(fact_582_zdvd__mult__cancel) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 206 (all B_93 all A_114 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_114)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_93)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_114),B_93)))))) # label(fact_1884_add__nonneg__pos) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 207 (all Y_1 all X_1 (-hBOOL(hAPP_int_bool(even_odd_even_int,X_1)) -> (-hBOOL(hAPP_int_bool(even_odd_even_int,Y_1)) -> hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Y_1)))))) # label(fact_2889_Parity_Oodd__plus__odd) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 208 (all A_228 A_228 = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_228),one_one_complex)) # label(fact_331_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 209 (all C all D all A all B (hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A),C)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B),D)) != hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A),D)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B),C)) <-> C != D & B != A)) # label(fact_316_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 210 (all N_11 all A_6 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_6)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_6),one_one_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_6),hAPP_nat_nat(suc,N_11))),one_one_int))))) # label(fact_3167_power__Suc__less__one) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 211 (all I all M hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),I))))) # label(fact_3057_less__add__Suc2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 212 (all A_228 (is_int(A_228) -> A_228 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_228),one_one_int))) # label(fact_327_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 213 (all Y_1 all X_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3345_real__sqrt__sum__squares__ge2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 214 (all Ma all V (hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,V)) = Ma & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V))) <-> hAPP_int_int(number_number_of_int,V) = hAPP_nat_int(semiri1621563631at_int,Ma))) # label(fact_3171_int__eq__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 215 (all C_41 all A_98 all B_81 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_41),A_98)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_41),B_81))) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_98),B_81)))) # label(fact_2036_add__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 216 (all X_2 all Y_2 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_2),Y_2)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_2)),Y_2))) # label(fact_13_power2__sum) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 217 (all X_2 hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,fact_fact_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_2)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_4839_exp__tail__after__first__two__terms__summable) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 218 (all M_11 -hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(semiri1424489471de_int,M_11)),zero_z891286103de_int))) # label(fact_2919_of__nat__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 219 (all N_56 all A_206 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_206)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_206),N_56))))) # label(fact_651_zero__le__power) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 220 (all M_25 all N_65 all A_216 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_216)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_216),M_25)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_216),N_65))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_25),N_65))))) # label(fact_529_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 221 (all X_2 all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(number_number_of_int,X_2)),hAPP_int_int(number_number_of_int,Y_2))) <-> hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,Y_2)),hAPP_int_int(number_number_of_int,X_2)) = zero_zero_int)) # label(fact_2443_dvd__eq__mod__eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 222 (all X_1 all Z_1 all D_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),Z_1))),one_one_int)),D_3))),Z_1)))) # label(fact_3463_decr__lemma) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 223 (all B_2 all A_3 all N_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_1)) -> (hAPP_int_int(legacy_zgcd(A_3),N_1) = one_one_int -> (exists X (is_int(X) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),X)),B_2),N_1)) & (all Y (is_int(Y) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),N_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),Y)),B_2),N_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y)) -> Y = X))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),N_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))))))) # label(fact_4532_zcong__lineq__unique) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 224 (all X_2 all Y_2 (zero_zero_real = X_2 & Y_2 = zero_zero_real <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_2),X_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_2),Y_2))),zero_zero_real)))) # label(fact_732_sum__squares__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 225 (all A_1 all B_1 (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,image_nat_int(semiri1621563631at_int,A_1)),image_nat_int(semiri1621563631at_int,B_1))) <-> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),B_1)))) # label(fact_4599_transfer__nat__int__set__relations_I5_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 226 (all X_1 all A_3 all B_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),B_2))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),B_2))))) # label(fact_4408_powr__less__cancel) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 227 (all N hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))),suc)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N))) = zero_zero_real) # label(fact_4910_sumr__minus__one__realpow__zero) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 228 (all A_70 all B_62 all C_30 hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_70),B_62)),C_30) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(div_mod_nat(A_70),C_30)),hAPP_nat_nat(div_mod_nat(B_62),C_30))),C_30)) # label(fact_2323_mod__mult__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 229 (all N (N = zero_zero_nat <-> zero_zero_real = hAPP_nat_real(real_nat,N))) # label(fact_3644_real__of__nat__zero__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 230 (all A_3 all B_2 all C_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B_2),C_1))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),hAPP_nat_nat(div_mod_nat(B_2),C_1))),C_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2)),C_1)) # label(fact_2743_div__mult1__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 231 (all B_14 all A_17 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_14),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_17),B_14))),hAPP_nat_nat(div_mod_nat(A_17),B_14)) = A_17) # label(fact_2672_mod__div__equality2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 232 (all P all Q all R hAPP_real_bool(hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,P),Q),R) = hAPP_real_bool(hAPP_r1134773055l_bool(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__HOL__Obo) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 233 (all D all Sa exists Z ((all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),Z)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Sa))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Sa)))))) & is_int(Z))) # label(fact_2255_minf_I9_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 234 (all A_3 all J_2 all K_2 all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),K_2),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),J_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),K_2))),P_3))))) # label(fact_2143_MultInv__zcong__prop1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 235 (all Y_1 all X_1 (hAPP_real_real(cos,X_1) != zero_zero_real -> (zero_zero_real != hAPP_real_real(cos,Y_1) -> (hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_1)) != zero_zero_real -> hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(tan,X_1)),hAPP_real_real(tan,Y_1))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(tan,X_1)),hAPP_real_real(tan,Y_1)))) = hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_1)))))) # label(fact_3611_tan__add) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 236 (all C_1 all A_3 all B_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(B_2),C_1),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),C_1),M))))) # label(fact_1005_zcong__trans) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 237 (all A_227 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),A_227) = A_227) # label(fact_335_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 238 (all K_2 all L_2 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),L_2)) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,K_2),L_2) = hAPP_int_int(z3div(K_2),L_2)) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),L_2)) -> hAPP_int_int(z3div(K_2),L_2) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,K_2),hAPP_int_int(uminus_uminus_int,L_2)))))) # label(fact_3949_z3div__def) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 239 (all A_181 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_181),A_181))) # label(fact_980_dvd__refl) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 240 (all Ma all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,Ma)),N)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)))) # label(fact_3064_Suc__le__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 241 (all A_3 all B_2 all C_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,B_2),C_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),C_1)) # label(fact_4310_gcd__assoc__nat) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 242 (all X_1 all Y_1 hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_1),Y_1)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_1))) # label(fact_3003_Nat__Transfer_Otransfer__int__nat__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 243 (all A_11 all N_18 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_11),hAPP_nat_nat(suc,N_18)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_11),N_18)),A_11)) # label(fact_3029_power__Suc2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 244 (all A_193 all M_21 all N_45 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_193),M_21)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_193),N_45)) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_193),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_21),N_45))) # label(fact_819_power__add) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 245 (all B_2 all A_3 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_2),A_3)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),A_3) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_2),A_3)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_3365_real__average__minus__second) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 246 (all M_13 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,one_on1645066479umeral),hAPP_n1108039445umeral(semiri1619134803umeral,M_13)) = hAPP_n1108039445umeral(semiri1619134803umeral,hAPP_nat_nat(suc,M_13))) # label(fact_2843_of__nat__Suc) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 247 (all V_2 hAPP_complex_real(im,hAPP_int_complex(number528085621omplex,V_2)) = zero_zero_real) # label(fact_4098_complex__Im__number__of) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 248 (all I all K_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,I)),K_2)) -> -(all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_1)) -> K_2 != hAPP_nat_nat(suc,J_1))))) # label(fact_3293_Suc__lessE) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 249 (all A_45 all C_13 all B_38 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_45),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_13),B_38))),B_38) = hAPP_int_int(div_mod_int(A_45),B_38)) # label(fact_2423_mod__mult__self1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 250 (all X_1 all Y_1 hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(cnj,X_1)),hAPP_complex_complex(cnj,Y_1)) = hAPP_complex_complex(cnj,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_1),Y_1))) # label(fact_4076_cnj_Odiff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 251 (all X_2 all Y_2 (hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2)) & hBOOL(hAPP_nat_bool(even_odd_even_nat,Y_2)) | -hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2)) & -hBOOL(hAPP_nat_bool(even_odd_even_nat,Y_2)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),Y_2)) <-> hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,X_2),Y_2))))) # label(fact_3763_even__difference__nat) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 252 (all N_59 all A_212 (is_int(A_212) -> (A_212 != zero_zero_int -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_212),N_59) != zero_zero_int))) # label(fact_590_field__power__not__zero) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 253 (all A_34 (is_int(A_34) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_34),one_one_int) = A_34)) # label(fact_2545_div__by__1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 254 (all B_96 all A_117 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_117),zero_z891286103de_int)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_96),zero_z891286103de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_117),B_96)),zero_z891286103de_int))))) # label(fact_1866_add__neg__nonpos) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 255 (all X_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) -> (hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_int_nat(nat,X_2))) <-> hBOOL(hAPP_int_bool(even_odd_even_int,X_2))))) # label(fact_3781_pos__int__even__equiv__nat__even) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 256 (all F all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) -> ((all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),B)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X)) -> (exists Y (hBOOL(hAPP_real_bool(deriv_real(F,X),Y)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),zero_zero_real)))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(F,B)),hAPP_real_real(F,A)))))) # label(fact_4548_DERIV__nonpos__imp__nonincreasing) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 257 (all A_8 all N_14 all B_7 (hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_8),hAPP_nat_nat(suc,N_14)) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_7),hAPP_nat_nat(suc,N_14)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_8)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_7)) -> A_8 = B_7)))) # label(fact_3113_power__inject__base) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 258 (all A all E_2 all C all B all D (hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A),E_2)),C) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B),E_2)),D) <-> C = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,B),A)),E_2)),D))) # label(fact_1549_eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 259 (all A_3 hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(number_number_of_int,min)) = hAPP_int_int(uminus_uminus_int,A_3)) # label(fact_3545_zdiv__minus1__right) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 260 (all F (hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,abs_abs_real),F))) -> hBOOL(summable_real(F)))) # label(fact_4827_summable__rabs__cancel) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 261 (all B all Q_1 all R_2 hAPP_P1975530577nt_int(adjust(B),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_1),R_2)) = hAPP_P1975530577nt_int(hAPP_P1145851913nt_int(hAPP_b40753821nt_int(if_Pro1731782967nt_int,hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,R_2),B))),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Q_1)),one_one_int)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,R_2),B))),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Q_1)),R_2))) # label(fact_4729_adjust__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 262 (all X_2 all Y_2 all Z_2 (Y_2 = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Z_2) <-> hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_2),Z_2) = X_2)) # label(fact_2703_eq__diff__eq_H) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 263 (all X_1 hAPP_real_real(tan,X_1) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sin,X_1)),hAPP_real_real(cos,X_1))) # label(fact_3672_tan__def) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 264 (all B all A all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int))) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(wset(A),P_5))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int))))))) # label(fact_2238_wset__less) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 265 (all B_94 all C_52 all A_115 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_115)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_94),C_52)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_94),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_115),C_52)))))) # label(fact_1878_add__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 266 (all A_171 all B_146 all C_84 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_171),B_146)),C_84)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_171),C_84)))) # label(fact_1096_dvd__mult__left) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 267 (all C_71 all B_123 all A_146 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_123),A_146)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_71),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_146),C_71)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_123),C_71)))))) # label(fact_1258_mult__strict__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 268 (all R_2 all X_3 (hBOOL(vanishes(X_3)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),R_2)) -> (exists K all N_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N_2)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,hAPP_nat_rat(X_3,N_2))),R_2))))))) # label(fact_5150_vanishesD) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 269 (all A_3 all P_3 (is_int(A_3) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),P_3)) -> hAPP_int_int(inv(P_3),hAPP_int_int(inv(P_3),A_3)) = A_3)))))) # label(fact_1892_inv__inv) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 270 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),posDivAlg(A_3,B_2)))))) # label(fact_2800_posDivAlg__correct) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 271 (all C_43 all A_100 all B_83 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_100),B_83)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_43),A_100)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_43),B_83))))) # label(fact_2013_add__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 272 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),zero_zero_int))))) # label(fact_2557_div__neg__pos__less0) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 273 (all D all Sa exists Z all X (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,X),Z)) -> (-hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,D),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X),Sa))) <-> -hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,D),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X),Sa)))))) # label(fact_2263_minf_I10_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 274 (all A_227 A_227 = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,one_on1645066479umeral),A_227)) # label(fact_337_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 275 (all A_166 all B_141 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_166),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_141),A_166)))) # label(fact_1128_dvd__triv__right) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 276 (all X_2 all A_1 (hBOOL(hAPP_P880235623l_bool(A_1,X_2)) <-> hBOOL(hAPP_f1469637905l_bool(hAPP_P1681069257l_bool(member308914164t_bool,X_2),A_1)))) # label(fact_205_mem__def) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 277 (all P all Q all R hAPP_i627165055t_real(hAPP_f1323932638t_real(hAPP_f352226557t_real(cOMBC_824100275t_real,P),Q),R) = hAPP_f269654879t_real(hAPP_i1671359920t_real(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__RealDef__Or) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 278 (all X_1 all Y_1 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_fst_int_int,bezw(X_1,Y_1))),hAPP_nat_int(semiri1621563631at_int,X_1))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_snd_int_int,bezw(X_1,Y_1))),hAPP_nat_int(semiri1621563631at_int,Y_1))) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Y_1))) # label(fact_4296_bezw__aux) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 279 (all P_5 hBOOL(hAPP_f448129468l_bool(resSet(P_5),hAPP_i1948725293t_bool(sRStar,P_5)))) # label(fact_4054_ResSet__SRStar__prop) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 280 (all A_61 all B_53 all C_24 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_61),B_53)),C_24) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(div_mod_int(A_61),C_24)),B_53)),C_24)) # label(fact_2354_mod__add__left__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 281 (all N hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),N)))))) # label(fact_4770_bdd__int__set__l__l__finite) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 282 (all F (hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,abs_abs_real),F))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_f352196356l_real(suminf_real,F))),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,abs_abs_real),F)))))) # label(fact_4829_summable__rabs) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 283 (all X_38 all Y_30 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_38),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_30),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Y_30)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_38),Y_30))))) # label(fact_788_power2__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 284 (all Lx_4 all Ly_2 all Rx_4 all Ry_2 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_4),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ly_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_4),Ry_2))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_4),Ly_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_4),Ry_2))) # label(fact_151_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 285 (all C_41 all A_98 all B_81 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_41),A_98)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_41),B_81))) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_98),B_81)))) # label(fact_2041_add__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 286 (all A_3 all M all B_2 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),M)),M))) # label(fact_1165_zcong__zmult__self) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 287 (all X_2 all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,X_2)),hAPP_nat_int(semiri1621563631at_int,Y_2))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),Y_2)))) # label(fact_2991_Nat__Transfer_Otransfer__int__nat__relations_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 288 (all X_35 all Y_27 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_35),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_27),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_847_sum__power2__ge__zero) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 289 (all A_3 hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(cos,A_3)),hAPP_real_real(sin,A_3)) = cis(A_3)) # label(fact_4151_cis__def) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 290 (all X_1 all Y_1 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)))) # label(fact_1709_dvd_Oless__imp__le) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 291 (all B_49 all B_48 all A_57 all C_20 all A_56 (hAPP_nat_nat(div_mod_nat(A_56),C_20) = hAPP_nat_nat(div_mod_nat(A_57),C_20) -> (hAPP_nat_nat(div_mod_nat(B_48),C_20) = hAPP_nat_nat(div_mod_nat(B_49),C_20) -> hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_57),B_49)),C_20) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_56),B_48)),C_20)))) # label(fact_2371_mod__add__cong) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 292 (all J all A all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),zero_zero_int),P_5)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_5),A)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J),zero_zero_int),P_5)) -> hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(multInvPair(A,P_5),J)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3289_MultInvPair__card__two) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 293 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) -> zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1))) # label(fact_1662_diff__is__0__eq_H) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 294 (all N_57 all A_211 all B_157 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_211),B_157)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_211)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_57)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_211),N_57)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_157),N_57))))))) # label(fact_603_power__strict__mono) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 295 (all K_2 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),K_2) = hAPP_int_int(bit0,K_2)) # label(fact_360_Bit0__def) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 296 (all Y_1 all X_1 (hBOOL(hAPP_int_bool(twoSqu1770640450sum2sq,X_1)) -> (hBOOL(hAPP_int_bool(twoSqu1770640450sum2sq,Y_1)) -> hBOOL(hAPP_int_bool(twoSqu1770640450sum2sq,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_1)))))) # label(fact_135_is__mult__sum2sq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 297 (all X_29 all Y_23 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_29),Y_23)) -> (Y_23 != X_29 -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_29),Y_23))))) # label(fact_1013_order__le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 298 (all A_18 all B_15 (is_int(A_18) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(div_mod_int(A_18),B_15)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_18),B_15)),B_15)) = A_18)) # label(fact_2667_semiring__div__class_Omod__div__equality_H) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 299 (all M_13 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,one_one_complex),hAPP_nat_complex(semiri2020571505omplex,M_13)) = hAPP_nat_complex(semiri2020571505omplex,hAPP_nat_nat(suc,M_13))) # label(fact_2844_of__nat__Suc) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 300 (all Y_1 all X_1 (hBOOL(hAPP_int_bool(nat_is_nat,X_1)) -> (hBOOL(hAPP_int_bool(nat_is_nat,Y_1)) -> hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(div_mod_int(X_1),Y_1)))))) # label(fact_5135_Divides_Otransfer__int__nat__function__closures_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 301 (all A all N all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A),hAPP_nat_nat(suc,N))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B),hAPP_nat_nat(suc,N)))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A),B)))) # label(fact_4029_coprime__exp2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 302 (all A all E_2 all C all B all D (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,B),A)),E_2)),D))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),E_2)),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),E_2)),D))))) # label(fact_1571_le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 303 (all A all P_5 hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(wset(A),P_5)))) # label(fact_3356_wset__fin) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 304 (all A_112 all B_91 all C_51 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_112),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_91),C_51)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_112),B_91)),C_51)) # label(fact_1905_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 305 (all N (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N)),zero_zero_real)) <-> N = zero_zero_nat)) # label(fact_3658_real__of__nat__le__zero__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 306 (all A_192 all M_20 all N_43 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_192),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_20),N_43)) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_192),M_20)),N_43)) # label(fact_832_power__mult) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 307 (all P_5 hAPP_i1948725293t_bool(sRStar,P_5) = collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),P_5)))) # label(fact_2521_SRStar__def) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 308 (all Z_4 all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1)) -> hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),Z_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(nat,Z_1)),hAPP_int_nat(nat,Z_4)))) # label(fact_2755_nat__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 309 (all M all N_1 all K_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_1)))) # label(fact_2736_nat__mult__div__cancel1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 310 (all N_16 all M_5 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_16),M_5)) -> hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_5),N_16)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(semiri1621563631at_int,M_5)),hAPP_nat_int(semiri1621563631at_int,N_16)))) # label(fact_3104_of__nat__diff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 311 (all M all K_2 (M != hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K_2) -> (M != hAPP_nat_nat(suc,zero_zero_nat) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M))))) # label(fact_3132_one__less__m) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 312 (all A all X_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),archim1246769320r_real(X_2))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,A)),X_2)))) # label(fact_4416_le__floor__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 313 (all Lx_5 all Ly_3 all Rx_5 all Ry_3 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_5),Ly_3)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_5),Ry_3)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_5),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_5),Ly_3)),Ry_3))) # label(fact_147_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 314 (all P all Q all R hAPP_int_bool(P,hAPP_nat_int(Q,R)) = hAPP_nat_bool(hAPP_f1154658180t_bool(hAPP_f1361476881t_bool(cOMBB_int_bool_nat,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__HOL__Obool_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 315 (all A_3 all B_2 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_3),B_2)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),A_3) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_2),A_3)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_3364_real__average__minus__first) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 316 (all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,K_1)),min)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),min)))) # label(fact_941_rel__simps_I30_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 317 (all X_1 hAPP_real_real(sin,X_1) = hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),pi)))) # label(fact_3717_sin__periodic) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 318 (all A_74 A_74 = hAPP_Q1762011733de_int(div_mo231679042de_int(A_74),zero_z891286103de_int)) # label(fact_2309_mod__by__0) # label(axiom) # label(non_clause). [assumption]. 5.34/5.39 319 (all N_44 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,one_one_real),N_44) = one_one_real) # label(fact_827_power__one) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 320 (all N all G all X_2 all Ma (hBOOL(hAPP_real_bool(deriv_real(G,X_2),Ma)) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),G)),N),X_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(G,X_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)))),Ma))))) # label(fact_4772_DERIV__fun__pow) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 321 (all Z_10 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_10),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z_10),Z_10)) # label(fact_498_semiring__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 322 (all X_2 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zOdd)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),one_one_int)),zEven)))) # label(fact_3153_odd__minus__one__even) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 323 (all A all B all C all D (hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),D) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B) -> (B = A <-> C = D))) # label(fact_1960_diff__eq__diff__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 324 (all C all A all B (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A),B)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C),A)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C),B))))) # label(fact_2071_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 325 (all C_72 all B_124 all A_147 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_124),A_147)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_72),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_72),A_147)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_72),B_124)))))) # label(fact_1254_mult__strict__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 326 (all A_179 all B_151 (hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_179),B_151) = zero_z891286103de_int -> zero_z891286103de_int = A_179 | zero_z891286103de_int = B_151)) # label(fact_1023_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 327 (all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N_1)) & -hBOOL(hAPP_nat_bool(prime,N_1)) -> (exists M_1 exists K (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K),N_1)) & N_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),K) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),K)))))) # label(fact_4207_not__prime__ex__mk) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 328 (all B_74 all A_85 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_85),zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_74),zero_zero_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_85),B_74)),zero_zero_nat))))) # label(fact_2150_add__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 329 (all P all Q (hBOOL(hAPP_bool_bool(fimplies(P),Q)) | -hBOOL(Q))) # label(help_fimplies_2_1_U) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 330 (all A (is_int(A) -> (A = zero_zero_int <-> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),A) = zero_zero_int))) # label(fact_1977_double__zero__sym) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 331 (all P all Q all R hAPP_f237669397t_bool(P,hAPP_P989584703t_bool(Q,R)) = hAPP_P989584703t_bool(hAPP_f1781059817t_bool(hAPP_f326728547t_bool(cOMBB_2086486755nt_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__H) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 332 (all P all Q all R hAPP_f800361423l_real(P,hAPP_r2019696015l_real(Q,R)) = hAPP_r712953929l_real(hAPP_f1119546819l_real(hAPP_f555557763l_real(cOMBB_422423973l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__prod_Itc__RealDef__Orea_041) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 333 (all A_1 (hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_nat_int(semiri1621563631at_int,A_1))) <-> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1)))) # label(fact_4605_transfer__nat__int__set__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 334 (all A_29 all B_25 all C_10 (zero_z126310315umeral != C_10 -> hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_29),B_25) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_10),A_29)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_10),B_25)))) # label(fact_2583_div__mult__mult1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 335 (all Z_2 hilbert_Eps_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_f1402056515l_bool(hAPP_f824790735l_bool(cOMBB_real_bool_real,hAPP_r1134773055l_bool(fequal_real,hAPP_complex_real(re,hAPP_complex_complex(sgn_sgn_complex,Z_2)))),cos))),hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_f1402056515l_bool(hAPP_f824790735l_bool(cOMBB_real_bool_real,hAPP_r1134773055l_bool(fequal_real,hAPP_complex_real(im,hAPP_complex_complex(sgn_sgn_complex,Z_2)))),sin))),hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,pi)))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,ord_less_eq_real),pi))))) = arg(Z_2)) # label(fact_4807_arg__def) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 336 (all N_1 all M ((zero_zero_nat = M -> hAPP_nat_nat(fact_fact_nat,N_1) = hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1))) & (zero_zero_nat != M -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1)),hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N_1))) = hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1))))) # label(fact_3798_fact__add__num__eq__if2__nat) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 337 (all B_2 all A_3 all A_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),A_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_5),B_2)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)))))) # label(fact_2622_zdiv__mono1__neg) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 338 (all B_95 all C_53 all A_116 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_116)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_95),C_53)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_95),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_116),C_53)))))) # label(fact_1871_add__strict__increasing2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 339 (all X_47 all P_8 all Q_12 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_47),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_8),Q_12)) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_47),P_8)),Q_12)) # label(fact_69_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 340 (all A_71 all B_63 all C_31 hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_71),B_63)),C_31) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(div_mod_nat(A_71),C_31)),B_63)),C_31)) # label(fact_2319_mod__mult__left__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 341 (all X_2 all Y_2 (X_2 = Y_2 <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_2)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2)))) # label(fact_1691_dvd_Oeq__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 342 (all X_1 all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,X_1)),hAPP_real_real(sqrt,Y_1))))) # label(fact_3316_real__sqrt__less__mono) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 343 (all N_1 hAPP_nat_int(semiri1621563631at_int,N_1) = archim1246769320r_real(hAPP_nat_real(real_nat,N_1))) # label(fact_4019_floor__real__of__nat) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 344 (all P all Q all R hAPP_nat_complex(hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,P),Q),R) = hAPP_complex_complex(P,hAPP_nat_complex(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Complex__Ocomplex_000tc__Complex__Ocomplex_000tc__Na) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 345 (all N all Ma (is_int(Ma) -> (zero_zero_int != Ma -> (one_one_int = hAPP_int_int(abs_abs_int,N) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ma),N)),Ma)))))) # label(fact_3448_zdvd__mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 346 (all X_48 all N_69 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_48),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_69)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_48),N_69)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_48),N_69))) # label(fact_44_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 347 (all N_55 all A_205 all B_156 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_205),B_156)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_205)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_205),N_55)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_156),N_55)))))) # label(fact_658_power__mono) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 348 (all X_1 all A_3 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3)))) # label(fact_4405_powr__gt__zero) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 349 (all L all U hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_at4362885an_nat(L),U)))) # label(fact_4895_finite__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 350 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),negDivAlg(A_3,B_2)))))) # label(fact_2796_negDivAlg__correct) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 351 (all X_37 all Y_29 (is_int(X_37) & is_int(Y_29) -> (hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_37),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_29),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_37)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_29)) -> Y_29 = X_37))))) # label(fact_789_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 352 (all X_1 all Y_1 hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_real_real(uminus_uminus_real,Y_1))) = hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Y_1),hAPP_real_real(uminus_uminus_real,X_1)))) # label(fact_3487_abs__minus__add__cancel) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 353 (all A_192 all M_20 all N_43 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_192),M_20)),N_43) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_192),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_20),N_43))) # label(fact_838_power__mult) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 354 (all D all Sa exists Z all X (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z),X)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Sa))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Sa)))))) # label(fact_2250_pinf_I9_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 355 (all A_70 all B_62 all C_30 hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_70),C_30)),hAPP_Q1762011733de_int(div_mo231679042de_int(B_62),C_30))),C_30) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_70),B_62)),C_30)) # label(fact_2325_mod__mult__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 356 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K1)),hAPP_int_int(bit0,K2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)))) # label(fact_116_less__eq__int__code_I13_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 357 (all Ta all B_1 all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ta),B_1)) -> (all X (is_int(X) -> ((all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B_1)) -> X != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa))))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ta),X)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ta),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),D_1)))))))))) # label(fact_5097_bset_I7_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 358 (all X_1 hAPP_real_real(log(hAPP_real_real(exp_real,one_one_real)),X_1) = hAPP_real_real(ln,X_1)) # label(fact_3868_log__ln) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 359 (all N_1 all A_3 cis(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_1))),A_3)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,cis(A_3)),cis(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),A_3)))) # label(fact_4153_cis__real__of__nat__Suc__mult) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 360 (all X_2 (-hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2)) <-> (exists Y X_2 = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),Y))))) # label(fact_3769_odd__nat__equiv__def2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 361 (all B_96 all A_117 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_117),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_96),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_117),B_96)),zero_zero_real))))) # label(fact_1865_add__neg__nonpos) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 362 (all X_36 all Y_28 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_36),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_28),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_28)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_36),Y_28))))) # label(fact_841_power2__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 363 (all X_1 all Y_1 all Z_1 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y_1),Z_1)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),Y_1)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),Z_1))) # label(fact_104_zpower__zadd__distrib) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 364 (all A_76 all B_65 hAPP_int_int(div_mod_int(hAPP_int_int(div_mod_int(A_76),B_65)),B_65) = hAPP_int_int(div_mod_int(A_76),B_65)) # label(fact_2298_mod__mod__trivial) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 365 (all P (is_bool(P) -> fTrue = P | fFalse = P)) # label(help_If_3_1_If_000tc__Nat__Onat_T) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 366 (all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(suc,zero_zero_nat))) = N_1)) # label(fact_3136_Suc__pred) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 367 (all X_2 all N (-hBOOL(hAPP_nat_bool(even_odd_even_nat,N)) -> (X_2 != zero_zero_real -> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(root,N),X_2),hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,zero_zero_nat)))))))))) # label(fact_3940_DERIV__odd__real__root) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 368 (all P all Q all R hAPP_i872162435t_bool(hAPP_f1251097615t_bool(hAPP_f1475822739t_bool(cOMBB_2027861294ol_int,P),Q),R) = hAPP_f1512942609t_bool(P,hAPP_i1948725293t_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__025) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 369 (all M_13 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,one_on1684967323de_int),hAPP_n522471361de_int(semiri1424489471de_int,M_13)) = hAPP_n522471361de_int(semiri1424489471de_int,hAPP_nat_nat(suc,M_13))) # label(fact_2845_of__nat__Suc) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 370 (all Ma all X all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),Xa),Ma)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcongm(Ma),X),Xa)))) # label(fact_5152_zcongm__def) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 371 (all X_1 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sin,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3729_sin__squared__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 372 (all B_1_1 all B_2_1 is_bool(hAPP_P1555980039t_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 373 (all X_2 all P_1 ((all A_2 (is_int(A_2) -> ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_2)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_2),one_one_int)))) -> hBOOL(hAPP_int_bool(P_1,A_2))))) -> hBOOL(hAPP_int_bool(P_1,X_2)))) # label(fact_2232_d22set__induct__old) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 374 (all A_230 all M_29 all B_163 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_230),M_29)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_163),M_29)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_230),B_163)),M_29)) # label(fact_303_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 375 (all A_148 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_148),A_148)))) # label(fact_1252_zero__le__square) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 376 (all V_9 all U_2 all Y_20 all X_26 all A_122 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_26),A_122)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_20),A_122)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),U_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),V_9)) -> (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,U_2),V_9) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,U_2),X_26)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_9),Y_20))),A_122)))))))) # label(fact_1490_convex__bound__le) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 377 (all M_13 hAPP_nat_nat(semiri984289939at_nat,hAPP_nat_nat(suc,M_13)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_nat_nat(semiri984289939at_nat,M_13))) # label(fact_2841_of__nat__Suc) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 378 (all A all B all C (hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A),B) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A),C) <-> C = B)) # label(fact_1942_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 379 (all A_93 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_93),zero_zero_real) = A_93) # label(fact_2079_diff__0__right) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 380 (all B_5 all Q_8 all R_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_5),Q_8)),R_5)),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_5)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_8),zero_zero_int)))))) # label(fact_1498_q__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 381 (all K_1 all L (is_int(K_1) & is_int(L) -> (L = K_1 <-> hAPP_int_int(bit0,K_1) = hAPP_int_int(bit0,L)))) # label(fact_259_rel__simps_I48_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 382 (all A all X_3 all X_2 (hBOOL(tendsto_nat_real(X_3,X_2,sequentially)) -> ((exists N_3 all N_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_3),N_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(X_3,N_2)),A)))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),A))))) # label(fact_5034_LIMSEQ__le__const2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 383 (all X_2 hBOOL(hAPP_real_bool(isCont_real_real(arctan),X_2))) # label(fact_5161_isCont__arctan) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 384 (all M all K_2 all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_2)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K_2)),hAPP_nat_nat(suc,J_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_2))))) # label(fact_3141_diff__Suc__diff__eq1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 385 (all P all Q all R hAPP_i1948725293t_bool(hAPP_i345030060t_bool(P,R),Q) = hAPP_i1948725293t_bool(hAPP_i345030060t_bool(hAPP_f791698359t_bool(cOMBC_241247961t_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Int__Oint_000tc__fun_Itc__Int__Oint) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 386 (all Y_1 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1)) -> hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(ln,X_1)),hAPP_real_real(ln,Y_1)) = hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_1),Y_1))))) # label(fact_3455_ln__div) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 387 (all X_1 all Y_1 hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_1)),hAPP_real_real(cos,Y_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X_1)),hAPP_real_real(sin,Y_1)))) # label(fact_3683_cos__add) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 388 (all Z_1 hBOOL(hAPP_int_bool(nat_is_nat,hAPP_nat_int(semiri1621563631at_int,Z_1)))) # label(fact_5142_Nat__Transfer_Otransfer__int__nat__function__closures_I9_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 389 (all N_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(fact_fact_int,N_1))))) # label(fact_4280_fact__gt__zero__int) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 390 (all X_1 hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),X_1)) = hAPP_real_real(abs_abs_real,X_1)) # label(fact_3382_real__sqrt__abs2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 391 (all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_int_real(number267125858f_real,Y_2))))) # label(fact_763_le__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 392 (all M ((zero_zero_nat != M -> hAPP_nat_nat(fact_fact_nat,M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) & (M = zero_zero_nat -> hAPP_nat_nat(fact_fact_nat,M) = one_one_nat))) # label(fact_3814_fact__num__eq__if__nat) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 393 (all X_2 all Y_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_2)) <-> X_2 = Y_2)) # label(fact_1515_divides__antisym) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 394 (all A_213 all N_61 all B_158 (hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_158),N_61) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_213),N_61) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_213)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_158)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_61)) -> A_213 = B_158))))) # label(fact_577_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 395 (all A_30 all B_26 all C_11 (zero_z891286103de_int != C_11 -> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_30),C_11)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_26),C_11)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_30),B_26))) # label(fact_2580_div__mult__mult2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 396 (all A_221 A_221 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,pls))),A_221)) # label(fact_453_mult__numeral__1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 397 (all X_1 all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),zero_zero_real)) -> (exists R_1 exists A_2 (X_1 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(cos,A_2)) & hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(sin,A_2)) = Y_1)))) # label(fact_3946_polar__ex2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 398 (all X_1 all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(multInv(P_3),hAPP_int_int(multInv(P_3),X_1))),X_1),P_3)))))) # label(fact_2145_MultInv__prop4) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 399 (all A_3 all B_2 hAPP_int_rat(hAPP_int_fun_int_rat(fract,B_2),A_3) = hAPP_rat_rat(inverse_inverse_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2))) # label(fact_4683_inverse__rat) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 400 (all A_3 all B_2 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),B_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),A_3)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),B_2)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) # label(fact_1623_zdiff__power3) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 401 (all X_2 (X_2 = one_on1684967323de_int <-> one_on1684967323de_int = X_2)) # label(fact_1957_one__reorient) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 402 (all X_2 ((exists N_2 (is_int(N_2) & hAPP_int_real(real_int,N_2) = X_2)) <-> X_2 = hAPP_int_real(real_int,archim1246769320r_real(X_2)))) # label(fact_4380_real__of__int__floor__cancel) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 403 (all X_1 all Y_1 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,X_1)),hAPP_int_real(real_int,Y_1)) = hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Y_1))) # label(fact_4392_real__of__int__add) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 404 (all N_25 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N_25)))) # label(fact_2911_of__nat__0__le__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 405 (all N all V (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))) & (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))) -> N = hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))) <-> hAPP_nat_nat(suc,N) = hAPP_int_nat(number_number_of_nat,V))) # label(fact_4777_Suc__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 406 (all P_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,P_5),zOdd)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_2692_Euler_Oaux__2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 407 (all X_23 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_23),one_one_complex)),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_23),one_one_complex)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_23),X_23)),one_one_complex)) # label(fact_1545_real__squared__diff__one__factored) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 408 (all X_21 all N_31 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31)) -> hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_21),N_31) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_21),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_31),one_one_nat))),X_21))) # label(fact_1739_realpow__minus__mult) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 409 (all A_232 all C_92 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_232),C_92) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_92),A_232)) # label(fact_246_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 410 (all N_44 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,one_one_rat),N_44) = one_one_rat) # label(fact_831_power__one) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 411 (all K_2 all L_2 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(bit0,K_2)),L_2) = hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),L_2))) # label(fact_356_mult__Bit0) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 412 (all X_1 (zero_zero_real != hAPP_real_real(cos,X_1) -> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(tan,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(cos,X_1))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3856_tan__sec) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 413 (all A_3 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,A_3)),X_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),archim1246769320r_real(X_1))))) # label(fact_4418_le__floor) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 414 (all B_92 all A_113 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_113)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_92)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_113),B_92)))))) # label(fact_1886_add__pos__nonneg) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 415 (all X_46 X_46 = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_46),one_one_nat)) # label(fact_77_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 416 (all P all Q all R hAPP_nat_bool(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,P),Q),R) = hAPP_nat_bool(hAPP_n1699378549t_bool(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Nat__Onat_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 417 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),pi)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(sin,X_1)))))) # label(fact_3657_sin__gt__zero__pi) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 418 (all A_238 all B_167 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_167),A_238) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_238),B_167)) # label(fact_185_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 419 (all A_45 all C_13 all B_38 hAPP_C498520661umeral(div_mo1740067990umeral(A_45),B_38) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_45),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_13),B_38))),B_38)) # label(fact_2425_mod__mult__self1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 420 (all N_1 all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N_1),M)) -> M = zero_zero_nat | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1)),M)) | M = N_1)) # label(fact_1506_divides__cases) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 421 (all X_43 all Y_35 all Q_10 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_43),Y_35)),Q_10) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_43),Q_10)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_35),Q_10))) # label(fact_342_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 422 (all A all A_1 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma)) -> (hBOOL(hAPP_int_bool(is_RRset(A_1),Ma)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),A_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),hAPP_int_int(rRset2norRR(A_1,Ma),A)),Ma)) & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(rRset2norRR(A_1,Ma),A)),hAPP_i1948725293t_bool(norRRset,Ma))))))) # label(fact_4590_RRset2norRR__correct) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 423 (all N_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))))) # label(fact_3850_inv__real__of__nat__fact__gt__zero) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 424 (all A_49 all B_41 hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_49),B_41)),B_41) = zero_z126310315umeral) # label(fact_2409_mod__mult__self2__is__0) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 425 (all X_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_nat_real(real_nat,natceiling(X_1))))) # label(fact_3655_real__natceiling__ge) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 426 (exists X (is_int(X) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) & (all Y (is_int(Y) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(s1),Y),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))) -> X = Y))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(s1),X),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))))) # label(fact_543__096EX_B_As_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_A1_A_G_A_091s1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 427 (all Ma all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(suc,Ma))) <-> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)))) # label(fact_2857_not__less__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 428 (all C_1 all A_3 all B_2 (B_2 = A_3 -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),C_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),C_1))))) # label(fact_1700_dvd_Oord__eq__le__trans) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 429 (all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),A_3)) -> -(hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),A_3)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2))))) # label(fact_1716_dvd_Oless__asym_H) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 430 (all G all F all X_2 all D (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D)) -> ((all Z (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_2))),D)) -> Z = hAPP_real_real(G,hAPP_real_real(F,Z)))) -> ((all Z (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_2))),D)) -> hBOOL(hAPP_real_bool(isCont_real_real(F),Z)))) -> hBOOL(hAPP_real_bool(isCont_real_real(G),hAPP_real_real(F,X_2))))))) # label(fact_5175_isCont__inv__fun) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 431 (all A_191 all N_42 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_191),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_42))))) # label(fact_858_zero__le__even__power_H) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 432 (all X_2 all Y_2 (X_2 = Y_2 <-> hAPP_complex_real(re,Y_2) = hAPP_complex_real(re,X_2) & hAPP_complex_real(im,Y_2) = hAPP_complex_real(im,X_2))) # label(fact_4115_complex__eq__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 433 (all M_19 all N_38 all A_188 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_188)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_188),M_19)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_188),N_38))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_19),N_38))))) # label(fact_889_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 434 (all M all N_1 all Q_2 all R_3 (hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Q_2),R_3))) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N_1) = Q_2)) # label(fact_4217_div__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 435 (all Diff all F all N all H (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H),zero_zero_real)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hAPP_n546711566l_real(Diff,zero_zero_nat) = F -> ((all M_1 all T (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,H),T)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),zero_zero_real)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N)) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),T),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T))))) -> (exists T (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H),T)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,T),zero_zero_real)) & hAPP_real_real(F,H) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,H))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),N)))))))))) # label(fact_4935_Maclaurin__minus) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 436 (all X_38 all Y_30 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_38),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_30),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),Y_30)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_38),Y_30))))) # label(fact_787_power2__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 437 (all X hAPP_real_real(cos,X) = hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),cos_coeff)),hAPP_r474017924t_real(power_power_real,X)))) # label(fact_4821_lemma__cos__ext) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 438 (all X_1 hAPP_real_real(uminus_uminus_real,hAPP_complex_real(re,X_1)) = hAPP_complex_real(re,hAPP_complex_complex(uminus473333897omplex,X_1))) # label(fact_4123_Re_Ominus) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 439 (all X_1 all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(multInv(P_3),X_1)),X_1)),one_one_int),P_3)))))) # label(fact_2226_MultInv__prop2a) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 440 (all P all Q (hBOOL(fdisj(P,Q)) | -hBOOL(Q))) # label(help_fdisj_2_1_U) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 441 (all P (is_int(P) -> hAPP_int_int(cOMBI_int,P) = P)) # label(help_COMBI_1_1_COMBI_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 442 (all C all A all B (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),A)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),B))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C),zero_zero_rat)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B),A)) | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A),B)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C)))) # label(fact_1325_mult__less__cancel__left__disj) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 443 (all A_3 all B_2 (quotient_of(B_2) = quotient_of(A_3) -> B_2 = A_3)) # label(fact_4653_quotient__of__inject) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 444 (all R_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),R_3)) -> (exists N_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N_2),one_one_int)))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N_2)),R_3)) & is_int(N_2))))) # label(fact_4477_reals__Archimedean__6b__int) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 445 (all X_7 all Q_5 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_7),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_7),Q_5)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_7),hAPP_nat_nat(suc,Q_5))) # label(fact_3026_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 446 (all C_80 all A_164 all B_139 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_164),B_139)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_164),C_80)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_164),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_139),C_80)))))) # label(fact_1146_dvd__add) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 447 (all X_1 hAPP_complex_complex(uminus473333897omplex,X_1) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(uminus_uminus_real,hAPP_complex_real(re,X_1))),hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,X_1)))) # label(fact_4134_complex__minus__def) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 448 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),zero_zero_int)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),B))))) # label(fact_1248_zero__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 449 (all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Y_2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(number_number_of_int,Y_2))))) # label(fact_756_less__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 450 (all B_1_1 is_bool(trivial_limit_nat(B_1_1))) # label(gsy_c_Limits_Otrivial__limit_000tc__Nat__Onat) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 451 (all X_2 (hAPP_nat_nat(div_mod_nat(X_2),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) = zero_zero_nat <-> hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2)))) # label(fact_3775_even__nat__equiv__def) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 452 (all A_207 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_207),zero_z126310315umeral) = A_207) # label(fact_629_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 453 (all X_1 all N_1 all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> hAPP_real_real(hAPP_n546711566l_real(root,N_1),hAPP_real_real(hAPP_n546711566l_real(root,M),X_1)) = hAPP_real_real(hAPP_n546711566l_real(root,M),hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1))))) # label(fact_3896_real__root__commute) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 454 (all A all B (hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(negDivAlg_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A),B))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),zero_zero_int)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),B))) -> negDivAlg(A,B) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),B))) & (-(hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),zero_zero_int)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),B)))) -> hAPP_P1975530577nt_int(adjust(B),negDivAlg(A,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B))) = negDivAlg(A,B)))) # label(fact_2530_negDivAlg_Opsimps) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 455 (all M_12 hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),hAPP_n1108039445umeral(semiri1619134803umeral,M_12)))) # label(fact_2914_zero__le__imp__of__nat) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 456 (all M_27 all A_224 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M_27),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_224),M_27)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_224),one_one_int)),M_27)) # label(fact_399_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 457 (all K_2 (is_int(K_2) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,pls),K_2) = K_2)) # label(fact_358_add__Pls) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 458 (all B_1_1 is_int(code_int_of(B_1_1))) # label(gsy_c_Code__Numeral_Oint__of) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 459 (all A_27 all B_24 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_27),B_24)) -> B_24 = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_27),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,B_24),A_27)))) # label(fact_2590_dvd__mult__div__cancel) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 460 (all X_1 all N_1 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(semiri1621563631at_int,X_1)),N_1) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),N_1))) # label(fact_3082_Nat__Transfer_Otransfer__int__nat__functions_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 461 (all V_7 all B_99 all C_55 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_7)),B_99)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_7)),C_55)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_7)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_99),C_55))) # label(fact_1555_right__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 462 (all A (zero_zero_real = A <-> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_real)) # label(fact_777_zero__eq__power2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 463 (all X_2 all Y_2 (is_int(Y_2) & is_int(X_2) -> (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_int <-> zero_zero_int = X_2 & Y_2 = zero_zero_int))) # label(fact_801_sum__power2__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 464 (all V_17 all B_160 all C_90 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_17)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_160),C_90)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_17)),B_160)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_17)),C_90))) # label(fact_386_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 465 (all A_203 all N_53 all B_155 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_203),N_53)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_155),N_53))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_155)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_203),B_155))))) # label(fact_673_power__less__imp__less__base) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 466 (all X_46 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_46),one_one_nat) = X_46) # label(fact_82_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 467 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)),M))))) # label(fact_1659_diff__less) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 468 (all M all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(suc,M)),N_1)) # label(fact_2868_add__Suc__shift) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 469 (all I_1 all J image_nat_nat(suc,hAPP_n1699378549t_bool(ord_at4362885an_nat(I_1),J)) = hAPP_n1699378549t_bool(ord_at4362885an_nat(hAPP_nat_nat(suc,I_1)),hAPP_nat_nat(suc,J))) # label(fact_4906_image__Suc__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 470 (all X_1 all I hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(real_int,I)),pi))) = hAPP_real_real(tan,X_1)) # label(fact_4453_tan__periodic__int) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 471 (all X_1 all Y_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Y_1) = X_1)) # label(fact_4322_gcd__proj1__if__dvd__nat) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 472 (all X_2 all A_1 (hBOOL(hAPP_f526364711l_bool(hAPP_P1062890741l_bool(member2143287562nt_int,X_2),A_1)) <-> hBOOL(hAPP_P62333565t_bool(A_1,X_2)))) # label(fact_204_mem__def) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 473 (all A_109 all B_88 all C_48 (hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_109),C_48) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_109),B_88) -> B_88 = C_48)) # label(fact_1924_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 474 (all A_170 all B_145 all K_7 (A_170 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_145),K_7) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_145),A_170)))) # label(fact_1105_dvdI) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 475 (all A_43 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_43),zero_zero_real) = zero_zero_real) # label(fact_2474_mult_Ozero__right) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 476 (all A_30 all B_26 all C_11 (C_11 != zero_z126310315umeral -> hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_30),B_26) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_30),C_11)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_26),C_11)))) # label(fact_2579_div__mult__mult2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 477 (all B_2 all A_3 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M)) -> (one_one_int = hAPP_int_int(legacy_zgcd(A_3),M) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M)) -> hAPP_int_int(legacy_zgcd(B_2),M) = one_one_int)))) # label(fact_4522_zgcd__zcong__zgcd) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 478 (all P_1 hBOOL(hAPP_f448129468l_bool(nat_nat_set,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),P_1))))) # label(fact_4747_Nat__Transfer_Otransfer__int__nat__set__function__closures_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 479 (all B_2 all A_3 (is_int(A_3) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hAPP_P1975530577nt_int(negateSnd,posDivAlg(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2))) = divmod_int(A_3,B_2)) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> divmod_int(A_3,B_2) = negDivAlg(A_3,B_2))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_2)) -> posDivAlg(A_3,B_2) = divmod_int(A_3,B_2)) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_2)) -> (A_3 = zero_zero_int -> divmod_int(A_3,B_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),zero_zero_int)) & (zero_zero_int != A_3 -> hAPP_P1975530577nt_int(negateSnd,negDivAlg(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2))) = divmod_int(A_3,B_2)))))) # label(fact_4161_divmod__int__def) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 480 (all X_11 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_11)) -> hAPP_nat_real(semiri132038758t_real,hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_11))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(semiri132038758t_real,hAPP_int_nat(nat,X_11))),hAPP_nat_real(semiri132038758t_real,hAPP_int_nat(nat,X_11))))) # label(fact_2812_of__nat__double) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 481 (all B_114 all A_137 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_137)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_114),zero_z126310315umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_137),B_114)),zero_z126310315umeral))))) # label(fact_1311_mult__pos__neg) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 482 (all M all X_1 hAPP_int_int(standardRes(M),X_1) = hAPP_int_int(div_mod_int(X_1),M)) # label(fact_2296_StandardRes__def) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 483 (all X_24 all Y_18 all A_120 all B_100 hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_24),Y_18)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_120),B_100)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_24),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,Y_18),B_100))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_24),A_120)),B_100))) # label(fact_1541_mult__diff__mult) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 484 (all P all Q all R hAPP_i744251628nt_int(hAPP_f836602773nt_int(hAPP_f888249301nt_int(cOMBB_1875755114nt_int,P),Q),R) = hAPP_f747162069nt_int(P,hAPP_int_fun_int_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc___023) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 485 (all C all D all A all B (is_int(A) & is_int(B) & is_int(D) & is_int(C) -> (D != C & B != A <-> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),C)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),D)) != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),D)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),C))))) # label(fact_315_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 486 (all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N),zero_zero_int)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N)),zero_zero_real)))) # label(fact_4439_real__of__int__le__zero__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 487 (all A_31 all B_27 (is_int(A_31) & is_int(B_27) -> (zero_zero_int != B_27 -> A_31 = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_27),A_31)),B_27)))) # label(fact_2573_div__mult__self1__is__id) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 488 (all N hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(hAPP_b646336293l_real(if_real,hAPP_nat_bool(even_odd_even_nat,N)),zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,zero_zero_nat))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),N)) = zero_zero_real) # label(fact_3788_lemma__STAR__sin) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 489 (all A_39 all B_32 all B_31 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_39),B_32)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_39),B_31)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_39),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_32),B_31))) # label(fact_2486_mult_Odiff__right) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 490 (all Q_8 all R_5 all A_3 all B_2 all Q_2 all R_3 (is_int(Q_8) & is_int(Q_2) & is_int(B_2) -> (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3))) -> (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_8),R_5))) -> (B_2 != zero_zero_int -> Q_8 = Q_2))))) # label(fact_2790_unique__quotient) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 491 (all A_232 all C_92 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_92),A_232) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_232),C_92)) # label(fact_244_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 492 (all D all Sa exists Z all X (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X),Z)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,D),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X),Sa))) <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,D),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X),Sa)))))) # label(fact_2253_minf_I9_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 493 (all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_2772_Int2_Oaux__2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 494 (all K_1 all L (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,K_1)),hAPP_int_int(bit1,L))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),L)))) # label(fact_111_rel__simps_I34_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 495 (all X_1 -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(exp_real,X_1)),zero_zero_real))) # label(fact_3833_not__exp__less__zero) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 496 (all M all N_1 hAPP_int_int(div_mod_int(M),N_1) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,M),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,M),N_1)),N_1))) # label(fact_2641_zmod__zdiv__equality_H) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 497 (all A_61 all B_53 all C_24 hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_61),B_53)),C_24) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(div_mod_nat(A_61),C_24)),B_53)),C_24)) # label(fact_2355_mod__add__left__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 498 (all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> hAPP_nat_nat(fact_fact_nat,hAPP_int_nat(nat,X_1)) = hAPP_int_nat(nat,hAPP_int_int(fact_fact_int,X_1)))) # label(fact_4284_transfer__nat__int__factorial) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 499 (all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),P_3)))) # label(fact_4180_prime__g__one) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 500 (all M_6 all N_20 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_n522471361de_int(semiri1424489471de_int,M_6)),N_20) = hAPP_n522471361de_int(semiri1424489471de_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_6),N_20))) # label(fact_2979_of__nat__power) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 501 (all A_235 all B_164 all C_95 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_235),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_164),C_95)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_235),B_164)),C_95)) # label(fact_219_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 502 (all X_48 all N_69 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_48),N_69)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_48),N_69)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_48),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_69))) # label(fact_49_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 503 (all P all Q all R hAPP_int_int(P,hAPP_nat_int(Q,R)) = hAPP_nat_int(hAPP_f1139079189at_int(hAPP_f1431025877at_int(cOMBB_int_int_nat,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__Int__Oint_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 504 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(ln,X_1))))) # label(fact_3444_ln__gt__zero) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 505 (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X_2)),one_one_real)) -> hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,one_one_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_2)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))),zero_zero_real,sequentially)))) # label(fact_5049_zeroseq__arctan__series) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 506 (all X_30 all N_34 (X_30 = one_on1645066479umeral | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_34)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,X_30),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_30),N_34))))) # label(fact_993_dvd__power) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 507 (all X_1 all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,pi)),Y_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(cos,Y_1)),hAPP_real_real(cos,X_1))))))) # label(fact_3572_cos__monotone__minus__pi__0_H) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 508 (all X_1 all K_2 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sqrt,X_1)),K_2) = hAPP_real_real(sqrt,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),K_2))) # label(fact_3319_real__sqrt__power) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 509 (all V_4 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_4))) -> hAPP_nat_real(semiri132038758t_real,hAPP_int_nat(number_number_of_nat,V_4)) = hAPP_int_real(number267125858f_real,V_4)) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_4))) -> zero_zero_real = hAPP_nat_real(semiri132038758t_real,hAPP_int_nat(number_number_of_nat,V_4))))) # label(fact_3220_of__nat__number__of__lemma) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 510 (all X_2 all Y_2 all P_1 ((hBOOL(P_1) -> hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,P_1),X_2),Y_2)) = hAPP_int_nat(nat,X_2)) & (-hBOOL(P_1) -> hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,P_1),X_2),Y_2)) = hAPP_int_nat(nat,Y_2)))) # label(fact_2715_nat__if__cong) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 511 (all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (exists M_1 hAPP_nat_nat(suc,M_1) = N_1))) # label(fact_3264_gr0__implies__Suc) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 512 (all C all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),B)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C),A)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C),B))))) # label(fact_2025_add__le__cancel__left) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 513 (all U_1 all V_2 all X_1 all Y_1 (X_1 = Y_1 -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,U_1)),V_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),U_1)),Y_1))),V_2))))) # label(fact_3395_sin__bound__lemma) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 514 (all X_1 (hBOOL(hAPP_int_bool(nat_is_nat,X_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(fact_fact_int,X_1))))) # label(fact_5146_transfer__int__nat__factorial__closure) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 515 (all I (is_int(I) -> (I = zero_zero_int -> hAPP_int_int(sgn_sgn_int,I) = zero_zero_int) & (zero_zero_int != I -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),I)) -> hAPP_int_int(sgn_sgn_int,I) = hAPP_int_int(uminus_uminus_int,one_one_int)) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),I)) -> hAPP_int_int(sgn_sgn_int,I) = one_one_int)))) # label(fact_3601_zsgn__def) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 516 (all A all W_1 (A = zero_zero_nat & zero_zero_nat != hAPP_int_nat(number_number_of_nat,W_1) <-> zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A),hAPP_int_nat(number_number_of_nat,W_1)))) # label(fact_1383_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 517 (all C_42 all A_99 all B_82 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_99),B_82)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_99),C_42)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_82),C_42))))) # label(fact_2022_add__right__mono) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 518 (all X_1 all Y_1 all R_3 hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),R_3)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_1),R_3)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,X_1),Y_1)),of_real_complex(R_3))) # label(fact_3989_Complex__mult__complex__of__real) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 519 (all A_97 all C_40 all B_80 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_97),C_40)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_80),C_40))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_97),B_80)))) # label(fact_2044_add__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 5.34/5.40 520 (all M all N_1 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N_1)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1))) # label(fact_3004_int__mult) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 521 (all B_69 all A_80 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_80)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_69)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_80),B_69)))))) # label(fact_2195_add__pos__pos) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 522 (all X_45 all P_7 all Q_11 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_45),P_7)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_45),Q_11)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_45),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_11))) # label(fact_100_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 523 (all A_213 all N_61 all B_158 (hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_158),N_61) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_213),N_61) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_213)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_158)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_61)) -> B_158 = A_213))))) # label(fact_578_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 524 (all C_66 all D_13 all A_132 all B_111 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_132),B_111)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_66),D_13)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_111)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_66)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_132),C_66)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_111),D_13)))))))) # label(fact_1408_mult__strict__mono) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 525 (all K_2 all I all J_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),J_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),I)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),J_2))))) # label(fact_121_zadd__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 526 (all N_1 all M all A_3 all B_2 (hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2) = one_one_int -> one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N_1)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),M)))) # label(fact_4961_coprime__exp2__int) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 527 (all P all Q all R hAPP_i614188481l_bool(hAPP_f252743847l_bool(hAPP_f643658017l_bool(cOMBB_2046938128ol_int,P),Q),R) = hAPP_f1146629647l_bool(P,hAPP_i418383825t_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc_) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 528 (all Z_1 all X_1 all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z_1),X_1)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z_1),Y_1))))) # label(fact_1607_real__add__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 529 (all N_1 (one_one_nat = hAPP_nat_nat(div_mod_nat(N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) -> hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),one_one_nat)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_3791_lemma__even__mod__4__div__2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 530 (all A all C all B (zero_zero_real = C | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A),B)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),C)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),C))))) # label(fact_1378_dvd__mult__cancel__right) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 531 (all N_1 all M (is_int(M) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N_1),M)) -> M = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N_1),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,M),N_1))))) # label(fact_2563_zdvd__mult__div__cancel) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 532 (all X_2 (hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_2)))) # label(fact_3782_nat__even__iff__2__dvd) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 533 (all B_95 all C_53 all A_116 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_116)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_95),C_53)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_95),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_116),C_53)))))) # label(fact_1870_add__strict__increasing2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 534 (all N_1 all A_3 all B_2 all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),B_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N_1)),A_3)))))) # label(fact_4185_prime__power__dvd__cancel__right) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 535 (all X_1 all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),P_3)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(multInv(P_3),X_1)),zero_zero_int),P_3)))))) # label(fact_2146_MultInv__prop3) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 536 (all A_97 all C_40 all B_80 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_97),C_40)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_80),C_40))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_97),B_80)))) # label(fact_2045_add__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 537 (all Y_1 all X_1 all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> (one_one_real != A_3 -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1)) -> hAPP_real_real(log(A_3),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(log(A_3),X_1)),hAPP_real_real(log(A_3),Y_1))))))) # label(fact_3872_log__mult) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 538 (all X_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> hAPP_real_real(hAPP_n546711566l_real(root,N_1),hAPP_real_real(abs_abs_real,X_1)) = hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)))) # label(fact_3913_real__root__abs) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 539 (all X_1 all Y_1 all Z_1 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Y_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Z_1)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Y_1),Z_1))) # label(fact_264_zadd__left__commute) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 540 (all Ma all K_1 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),Ma)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),hAPP_nat_nat(div_mod_nat(Ma),N)))))) # label(fact_2393_dvd__mod__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 541 (all X_1 all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),X_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),P_3)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,X_1),P_3)))))) # label(fact_4186_prime__coprime__lt) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 542 (all R_2 hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(times_times_real,R_2)),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(plus_plus_real,one_one_real)),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,uminus_uminus_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),suc))))),R_2,sequentially))) # label(fact_5047_LIMSEQ__inverse__real__of__nat__add__minus__mult) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 543 (all X_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> hAPP_real_real(hAPP_n546711566l_real(root,N_1),hAPP_real_real(uminus_uminus_real,X_1)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)))) # label(fact_3912_real__root__minus) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 544 (all A all C all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),B)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A),C)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B),C))))) # label(fact_2074_add__less__cancel__right) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 545 (all C_1 all D_3 all A_3 all B_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C_1),D_3),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),C_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),D_3)),M))))) # label(fact_1160_zcong__zmult) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 546 (all K_2 all L_2 hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),L_2)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit0,K_2)),hAPP_int_int(bit0,L_2))) # label(fact_359_add__Bit0__Bit0) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 547 (all R_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),R_3)) -> (exists N_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N_2)),R_3)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_2)))))))) # label(fact_3749_reals__Archimedean6a) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 548 (all X_1 all A_3 hAPP_real_real(exp_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_3),hAPP_real_real(ln,X_1))) = hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3)) # label(fact_4436_powr__def) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 549 (all A_189 all N_40 all N_39 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_40),N_39)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),A_189)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_189),N_40)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_189),N_39)))))) # label(fact_867_power__increasing) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 550 (all P_3 (is_int(P_3) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) != P_3 -> (hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))) != P_3 -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),P_3))))))) # label(fact_512_prime__g__5) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 551 (all X_9 all Q_7 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_9),Q_7)),X_9) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_9),hAPP_nat_nat(suc,Q_7))) # label(fact_3011_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 552 (all B_97 all A_118 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_118),zero_z891286103de_int)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_97),zero_z891286103de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_118),B_97)),zero_z891286103de_int))))) # label(fact_1860_add__nonpos__neg) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 553 (all C all A all B (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C),A)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C),B))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A),B)))) # label(fact_2029_add__le__cancel__left) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 554 (all X_1 all Y_1 hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_1) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(uminus_uminus_int,X_1)),Y_1)) # label(fact_4992_gcd__neg1__int) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 555 (all N_1 all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),B_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_3),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_2),N_1))))) # label(fact_4018_coprime__exp__imp) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 556 (all M all N_1 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(semiri1621563631at_int,M)),N_1) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M),N_1))) # label(fact_3084_zpower__int) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 557 (all B_137 all A_161 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_161),zero_zero_int)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_137),zero_zero_int)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_161)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_137)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_161),B_137))))) # label(fact_1179_split__mult__pos__le) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 558 (all W_7 hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,W_7)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),one_one_real)),hAPP_int_real(number267125858f_real,W_7))) # label(fact_476_double__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 559 (all X_5 all N_8 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_5),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_8))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_5),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_5),N_8)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_5),N_8)))) # label(fact_3231_comm__semiring__1__class_Onormalizing__semiring__rules_I37_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 560 (all A all B all C all D (is_int(B) & is_int(D) & is_int(C) & is_int(A) -> (hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C),D) -> (B = A <-> D = C)))) # label(fact_1959_diff__eq__diff__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 561 (all X_2 (zero_zero_real = X_2 <-> one_one_real = hAPP_real_real(exp_real,X_2))) # label(fact_3835_exp__eq__one__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 562 (all K_2 hAPP_int_int(pred,K_2) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),one_one_int)) # label(fact_4044_pred__def) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 563 (all A_3 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))) -> hAPP_int_int(inv(P_3),A_3) != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))))) # label(fact_2130_inv__not__p__minus__1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 564 (all V_23 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_23)),one_one_real) = hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_23),hAPP_int_int(bit1,pls)))) # label(fact_55_add__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 565 (all N_1 all X_1 (is_int(N_1) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N_1)),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N_1)),one_one_real))) -> N_1 = archim1246769320r_real(X_1))))) # label(fact_4466_floor__eq2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 566 (all Nat_5 all Nat_4 (Nat_4 = Nat_5 <-> hAPP_nat_nat(suc,Nat_4) = hAPP_nat_nat(suc,Nat_5))) # label(fact_2837_nat_Oinject) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 567 (all A_148 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_148),A_148)))) # label(fact_1251_zero__le__square) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 568 (all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_int_real(number267125858f_real,Y_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,pls)),Y_2)))) # label(fact_288_less__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 569 (all X_2 all Y_2 (is_int(Y_2) & is_int(X_2) -> (X_2 = Y_2 <-> hAPP_int_real(real_int,Y_2) = hAPP_int_real(real_int,X_2)))) # label(fact_4383_real__of__int__inject) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 570 (all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) <-> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_n1108039445umeral(semiri1619134803umeral,N))))) # label(fact_3147_of__nat__0__less__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 571 (all M zero_zero_real = hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,pi),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),M))))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3630_cos__pi__eq__zero) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 572 (all C_64 all D_11 all A_130 all B_109 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_130),B_109)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_64),D_11)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_130)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_64)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_130),C_64)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_109),D_11)))))))) # label(fact_1420_mult__less__le__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 573 (all X_3 all A (hBOOL(tendsto_nat_complex(X_3,A,sequentially)) -> hBOOL(tendsto_nat_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,im),X_3),hAPP_complex_real(im,A),sequentially)))) # label(fact_5056_Im_OLIMSEQ) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 574 (all X_47 all P_8 all Q_12 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_47),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_8),Q_12)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_47),P_8)),Q_12)) # label(fact_74_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 575 (all N_28 all K_5 all M_15 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_5),M_15)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_5),N_28)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_5),hAPP_nat_nat(div_mod_nat(M_15),N_28)))))) # label(fact_2385_dvd__mod) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 576 (all M_25 all N_65 all A_216 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_216)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_216),M_25)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_216),N_65))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_25),N_65))))) # label(fact_533_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 577 (all X_2 all A (X_2 = A <-> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),hAPP_real_real(uminus_uminus_real,A)) = zero_zero_real)) # label(fact_3482_real__add__minus__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 578 (all A all E_2 all C all B all D (is_int(C) -> (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),E_2)),C) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),E_2)),D) <-> C = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B),A)),E_2)),D)))) # label(fact_1547_eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 579 (all B_40 all A_48 hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_40),A_48)),B_40) = zero_zero_nat) # label(fact_2412_mod__mult__self1__is__0) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 580 (all X_1 hAPP_real_real(tan,hAPP_real_real(uminus_uminus_real,X_1)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(tan,X_1))) # label(fact_3595_tan__minus) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 581 (all M all V_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,M)))),hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),M)),hAPP_int_nat(number_number_of_nat,V_2))) # label(fact_3206_Suc__div__eq__add3__div__number__of) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 582 (all X_2 all A all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),zero_zero_int),P_5)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),hAPP_i1948725293t_bool(sRStar,P_5))) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(standardRes(P_5),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),hAPP_int_int(multInv(P_5),X_2)))),hAPP_i1948725293t_bool(sRStar,P_5)))))))) # label(fact_2499_SRStar__mult__prop2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 583 (all P all Q all R hAPP_b40753821nt_int(P,hAPP_int_bool(Q,R)) = hAPP_i2104874254nt_int(hAPP_f251264923nt_int(hAPP_f279307923nt_int(cOMBB_662120866nt_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 584 (all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_real_real(sqrt,Y_2))))) # label(fact_3330_real__sqrt__ge__1__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 585 (all A_90 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_90),one_on1684967323de_int) = A_90) # label(fact_2112_mult__1__right) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 586 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)),hAPP_int_int(number_number_of_int,min))) -> hAPP_int_int(legendre(hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)) != one_one_int # label(fact_1507__096_126_AQuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_A_061_061_062_ALegendre_A_) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 587 (all W_1 all X_2 (X_2 = hAPP_int_real(number267125858f_real,W_1) <-> X_2 = hAPP_int_real(number267125858f_real,W_1))) # label(fact_253_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 588 (all A_59 all B_51 all C_22 hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_59),B_51)),C_22) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_59),hAPP_Q1762011733de_int(div_mo231679042de_int(B_51),C_22))),C_22)) # label(fact_2365_zmod__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 589 (all A all B (hAPP_Q1762011733de_int(div_mo231679042de_int(B),A) = zero_z891286103de_int <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A),B)))) # label(fact_2432_dvd__eq__mod__eq__0) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 590 (all Y_1 all X_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,Y_1)),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3406_real__sqrt__ge__abs2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 591 (all Y_21 all X_27 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),X_27)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Y_21)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_21),one_one_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_27),Y_21)),X_27)))))) # label(fact_1472_mult__right__le__one__le) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 592 (all B all A (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),zEven)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),zOdd)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),B)),zOdd))))) # label(fact_3096_IntNatAux_Oeven__plus__odd) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 593 (all X_2 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X_2)),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,fact_fact_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_2)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) = hAPP_real_real(exp_real,X_2)) # label(fact_4823_exp__first__two__terms) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 594 (all N all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),Ma))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)))) # label(fact_1658_zero__less__diff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 595 (all X_38 all Y_30 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_38),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_30),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_30)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_38),Y_30))))) # label(fact_784_power2__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 596 (all V_22 all V_21 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_21)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_22)) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_21)),hAPP_int_int(number_number_of_int,V_22)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_21),V_22))))) # label(fact_367_semiring__mult__number__of) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 597 (all X_2 all A_1 (hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X_2),A_1)) <-> hBOOL(hAPP_f448129468l_bool(A_1,X_2)))) # label(fact_201_mem__def) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 598 (all K_1 all L (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,K_1)),hAPP_int_int(number_number_of_int,L))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),L)))) # label(fact_119_less__eq__number__of__int__code) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 599 (all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),hAPP_nat_nat(suc,zero_zero_nat))) <-> hAPP_nat_nat(suc,zero_zero_nat) = Ma)) # label(fact_3052_dvd__1__iff__1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 600 (all N_1 all M all R_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),R_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N_1),R_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,M),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1)),R_3)))))) # label(fact_4025_divides__mul) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 601 (all N_1 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,N_1)),one_one_real) = hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_1))) # label(fact_3667_real__of__nat__Suc) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 602 (all A all B (B = A <-> hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A),B) = zero_zero_complex)) # label(fact_2092_right__minus__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 603 (all A_36 hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_36),zero_z126310315umeral) = zero_z126310315umeral) # label(fact_2539_div__by__0) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 604 (all A_127 all C_61 all B_106 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_127),C_61)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_106),C_61))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_61)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_127),B_106))))) # label(fact_1437_mult__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 605 (all N all N_4 ((all X (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),N_4)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),N)))) -> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_4)))) # label(fact_4626_bounded__nat__set__is__finite) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 606 (all A_3 all B_2 hAPP_int_int(div_mod_int(hAPP_nat_int(semiri1621563631at_int,A_3)),hAPP_nat_int(semiri1621563631at_int,B_2)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(div_mod_nat(A_3),B_2))) # label(fact_3087_zmod__int) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 607 (all A_3 all B_2 all P_3 all Q_2 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(twoSqu1907779896sum2sq,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_3),B_2))),hAPP_P1175774780nt_int(twoSqu1907779896sum2sq,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2))) = hAPP_P1175774780nt_int(twoSqu1907779896sum2sq,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),P_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_2))),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),Q_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),P_3))))) # label(fact_1608_mult__sum2sq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 608 (all N_33 one_one_complex = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_int_complex(number528085621omplex,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_33))) # label(fact_1171_power__m1__even) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 609 (all X hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,plus_plus_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,plus_plus_real),hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,sin),uminus_uminus_real))),sin))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,minus_minus_real),hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,cos),uminus_uminus_real))),cos))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X),zero_zero_real))) # label(fact_4802_lemma__DERIV__sin__cos__minus) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 610 (all V_2 all W ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W))) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,W)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,V_2))),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W)))) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W))) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_2)),one_one_int)),hAPP_int_int(number_number_of_int,W)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,V_2))),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W)))))) # label(fact_2702_zdiv__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 611 (all P all Q all R hAPP_i1948725293t_bool(P,hAPP_int_int(Q,R)) = hAPP_i1948725293t_bool(hAPP_f1791153283t_bool(hAPP_f1399575567t_bool(cOMBB_118231410ol_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 612 (all K_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(div_mod_int(M),K_2)),M)))) # label(fact_2437_zmod__le__nonneg__dividend) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 613 (all D all K_1 all P_5 (hBOOL(hAPP_nat_bool(prime,P_5)) -> ((exists I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),K_1)) & hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_5),I_2) = D)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_5),K_1)))))) # label(fact_4184_divides__primepow) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 614 (all Ma all N (hAPP_nat_nat(suc,zero_zero_nat) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),N) <-> Ma = zero_zero_nat & hAPP_nat_nat(suc,zero_zero_nat) = N | Ma = hAPP_nat_nat(suc,zero_zero_nat) & N = zero_zero_nat)) # label(fact_3053_one__is__add) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 615 (all Y_3 all Y_2 all X_3 all X_2 (hBOOL(tendsto_nat_real(X_3,X_2,sequentially)) -> (hBOOL(tendsto_nat_real(Y_3,Y_2,sequentially)) -> ((exists N_3 all N_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_3),N_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(X_3,N_2)),hAPP_nat_real(Y_3,N_2))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Y_2)))))) # label(fact_5035_LIMSEQ__le) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 616 (all B_90 all A_111 all C_50 (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_90),A_111) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_50),A_111) -> C_50 = B_90)) # label(fact_1913_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 617 (all X_2 all N all Y_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),Y_2)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_2),hAPP_nat_nat(suc,N))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_2),hAPP_nat_nat(suc,N)))))) # label(fact_3089_exp__mono__lt) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 618 (all C_69 all A_144 all B_121 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_144),B_121)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_69)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_69),A_144)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_69),B_121)))))) # label(fact_1268_mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 619 (all A_201 all N_50 all N_49 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_50),N_49)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_201)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_201),one_on1684967323de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_201),N_49)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_201),N_50))))))) # label(fact_685_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 620 (all X_1 all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),hAPP_int_int(multInv(P_3),X_1))),hAPP_int_int(multInv(P_3),hAPP_int_int(multInv(P_3),X_1)))),X_1),P_3)))))) # label(fact_2224_Int2_Oaux____2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 621 (all N all Ma (N = zero_zero_nat & zero_zero_nat = Ma <-> hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,Ma))) # label(fact_3529_negative__eq__positive) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 622 (all R_3 all A_3 hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),A_3)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),R_3)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,A_3),R_3)) # label(fact_3354_real__divide__square__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 623 (all A_13 all B_10 all Q_9 all Y_1 (is_int(Y_1) & is_int(B_10) -> (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_10),Q_9)),Y_1) = A_13 -> ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_10)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_1),zero_zero_int)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_10),Y_1))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_10)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),B_10))) -> (B_10 != zero_zero_int -> Y_1 = hAPP_int_int(div_mod_int(A_13),B_10)))))) # label(fact_2467_divmod__int__rel__mod__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 624 (all A_19 all B_16 (zero_zero_nat != B_16 -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_16),A_19)),B_16) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_19),B_16)),one_one_nat))) # label(fact_2656_div__add__self1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 625 (all N_32 all M_18 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),M_18)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),N_32)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M_18),N_32)))))) # label(fact_1362_less__1__mult) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 626 (all X_7 all Q_5 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_7),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_7),Q_5)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_7),hAPP_nat_nat(suc,Q_5))) # label(fact_3023_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 627 (all B_2 all A_3 (zero_zero_nat != A_3 | zero_zero_nat != B_2 -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2))) = one_one_nat)) # label(fact_4369_div__gcd__coprime__nat) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 628 (all Z_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natfloor(Z_1)),one_one_nat)),N_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z_1),hAPP_nat_real(real_nat,N_1))))) # label(fact_3703_ge__natfloor__plus__one__imp__gt) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 629 (all C_1 all A_3 all B_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),C_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_2),C_1)),M)))) # label(fact_1167_zcong__shift) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 630 (all A_45 all C_13 all B_38 hAPP_Q1762011733de_int(div_mo231679042de_int(A_45),B_38) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_45),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_13),B_38))),B_38)) # label(fact_2426_mod__mult__self1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 631 (all A_97 all C_40 all B_80 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_97),C_40)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_80),C_40))) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_97),B_80)))) # label(fact_2047_add__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 632 (all X_1 all N_1 hAPP_complex_complex(cnj,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_1),N_1)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_complex_complex(cnj,X_1)),N_1)) # label(fact_4077_complex__cnj__power) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 633 (all Z_1 hAPP_real_real(sqrt,hAPP_complex_real(re,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Z_1),hAPP_complex_complex(cnj,Z_1)))) = hAPP_complex_real(norm_norm_complex,Z_1)) # label(fact_4132_complex__mod__sqrt__Re__mult__cnj) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 634 (all V_22 all V_21 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_21)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_22)) -> hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_21),V_22)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_21)),hAPP_int_real(number267125858f_real,V_22))))) # label(fact_369_semiring__mult__number__of) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 635 (all Y_2 all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Y_2)) -> (exists X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),X)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y_2)) & hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X),field_1210416355s_real))))))) # label(fact_4633_Rats__dense__in__nn__real) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 636 (all A_173 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,zero_zero_real),A_173)) -> A_173 = zero_zero_real)) # label(fact_1078_dvd__0__left) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 637 (all A_73 hAPP_nat_nat(div_mod_nat(zero_zero_nat),A_73) = zero_zero_nat) # label(fact_2311_mod__0) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 638 (all Lx_1 all Rx_1 all Ry_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_1),Ry_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_1),Rx_1)),Ry_1)) # label(fact_173_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 639 (all X_1 all Y_1 hAPP_complex_real(im,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_complex_real(im,X_1)),hAPP_complex_real(im,Y_1))) # label(fact_4100_Im_Odiff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 640 (all B_112 all C_67 all A_134 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_134)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_112),C_67)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_112),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_134),C_67)))))) # label(fact_1334_pos__add__strict) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 641 (all K_2 all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),K_2))))) # label(fact_1761_add__less__mono1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 642 (all X_2 all Y_2 (Y_2 != zero_zero_rat | X_2 != zero_zero_rat <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_856_sum__power2__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 643 (all X_3 all A (hBOOL(hAPP_complex_bool(sums_complex(X_3),A)) -> hBOOL(hAPP_real_bool(sums_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,im),X_3)),hAPP_complex_real(im,A))))) # label(fact_4872_Im_Osums) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 644 (all F all Y_2 all X_2 all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)) -> (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),hAPP_r1134773055l_bool(ord_gr788844697n_real(A),B))) -> (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Y_2),hAPP_r1134773055l_bool(ord_gr788844697n_real(A),B))) -> ((all X (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X),hAPP_r1134773055l_bool(ord_gr788844697n_real(A),B))) -> hBOOL(hAPP_real_bool(deriv_real(F,X),zero_zero_real)))) -> hAPP_real_real(F,Y_2) = hAPP_real_real(F,X_2)))))) # label(fact_4634_DERIV__isconst3) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 645 (all X_1 (-hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1)) -> hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X_1),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))))) = X_1)) # label(fact_3780_odd__nat__div__two__times__two__plus__one) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 646 (all N_1 hAPP_nat_int(semiri1621563631at_int,N_1) = archim856651990g_real(hAPP_nat_real(real_nat,N_1))) # label(fact_4362_ceiling__real__of__nat) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 647 (all X_2 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(number_number_of_rat,X_2)),one_one_rat)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),hAPP_int_int(bit1,pls))))) # label(fact_286_less__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 648 (all N_56 all A_206 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_206)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_206),N_56))))) # label(fact_653_zero__le__power) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 649 (all Z_1 (is_int(Z_1) -> (-hBOOL(hAPP_int_bool(nat_neg,Z_1)) -> hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat,Z_1)) = Z_1))) # label(fact_3280_not__neg__nat) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 650 (all D all B all C all A (A != zero_z891286103de_int -> (C != zero_z891286103de_int -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A),B)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,C),D)) -> (hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,D),C) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,B),A) <-> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B),C) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A),D))))))) # label(fact_2660_dvd__div__div__eq__mult) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 651 (all A all D all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),D)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,hAPP_nat_fun_nat_nat(plus_plus_nat,A)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),D))),hAPP_n1699378549t_bool(ord_lessThan_nat,N)))) # label(fact_4952_arith__series__nat) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 652 (all A (hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially)) -> (hBOOL(monoseq_real(A)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(A,zero_zero_nat))) -> (all N_2 hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A))),hAPP_r1134773055l_bool(ord_at1589558736t_real(hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_2)))),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_2)),one_one_nat))))))))))) # label(fact_5062_summable__Leibniz_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 653 (all M_8 all N_22 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,M_8)),hAPP_nat_int(semiri1621563631at_int,N_22)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_8),N_22))) # label(fact_2961_of__nat__add) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 654 (all A_219 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_219),A_219)),A_219) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_219),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) # label(fact_482_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 655 (all R_3 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,R_3),one_one_real)),hAPP_int_real(real_int,archim1246769320r_real(R_3))))) # label(fact_4454_real__of__int__floor__ge__diff__one) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 656 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> hAPP_real_real(ln,hAPP_real_real(inverse_inverse_real,X_1)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(ln,X_1)))) # label(fact_3851_ln__inverse) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 657 (all G all F all X_2 all D (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D)) -> ((all Z (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_2))),D)) -> hAPP_real_real(G,hAPP_real_real(F,Z)) = Z)) -> ((all Z (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_2))),D)) -> hBOOL(hAPP_real_bool(isCont_real_real(F),Z)))) -> (exists Z (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,X_2)),hAPP_real_real(F,Z))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_2))),D)))))))) # label(fact_5181_lemma__isCont__inj) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 658 (all Y_1 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),Y_1)))))) # label(fact_1603_real__mult__order) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 659 (all A_3 all B_2 hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(uminus_uminus_real,B_2)),A_3) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),B_2))) # label(fact_3982_i__mult__Complex) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 660 (all A_3 all B_2 hAPP_complex_complex(cnj,hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),B_2)) = hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),hAPP_real_real(uminus_uminus_real,B_2))) # label(fact_4082_complex__cnj) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 661 (all N_27 all Y_8 all X_12 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Y_8),X_12)) -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_12),Y_8)),N_27) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_12),N_27)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_8),N_27)))) # label(fact_2619_div__power) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 662 (all A_19 all B_16 (is_int(B_16) -> (zero_zero_int != B_16 -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_16),A_19)),B_16) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_19),B_16)),one_one_int)))) # label(fact_2655_div__add__self1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 663 (all Z_2 all W_1 (is_int(Z_2) & is_int(W_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_2),W_1)) & Z_2 != W_1 <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_2),W_1))))) # label(fact_65_zless__le) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 664 (all X1_2 all X2 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X1_2),X2),Ma)) <-> hAPP_int_int(standardRes(Ma),X1_2) = hAPP_int_int(standardRes(Ma),X2)))) # label(fact_2460_StandardRes__prop2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 665 (all R_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),R_3)) -> (exists N_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),hAPP_nat_real(real_nat,N_2))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_2),one_one_nat))),R_3)))))) # label(fact_3747_reals__Archimedean6) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 666 (all X_1 (-hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1)) -> hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X_1),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,X_1)),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))))) # label(fact_3770_odd__nat__plus__one__div__two) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 667 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N_1))) -> (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) -> N_1 = M))) # label(fact_2865_less__SucE) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 668 (all R_3 all P_3 all Q_2 (quotient_of(R_3) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Q_2)))) # label(fact_4657_quotient__of__denom__pos) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 669 (all A_72 all B_64 all C_32 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_72),B_64)),C_32) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_72),hAPP_int_int(div_mod_int(B_64),C_32))),C_32)) # label(fact_2314_mod__mult__right__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 670 (all X_1 all Y_1 (Y_1 = X_1 | -hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(fequal_rat,X_1),Y_1)))) # label(help_fequal_1_1_fequal_000tc__Rat__Orat_T) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 671 (all K_2 all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),K_2))))) # label(fact_1777_add__le__mono1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 672 (all X_2 all A all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),zero_zero_int),P_5)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_5),A)) -> (hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X_2),setS(A,P_5))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(big_co1548731110nt_int(cOMBI_int,X_2)),A),P_5)))))))) # label(fact_4812_SetS__setprod__prop) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 673 (all N_1 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_1)))),zero_zero_int))) # label(fact_3547_negative__zless__0) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 674 (all X_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_complex_real(norm_norm_complex,X_1))),hAPP_complex_real(norm_norm_complex,X_1)))) # label(fact_3488_complex__mod__minus__le__complex__mod) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 675 (all P all Q all R hAPP_P1333854989l_bool(P,hAPP_r1195171167l_real(Q,R)) = hAPP_real_bool(hAPP_f330985925l_bool(hAPP_f940061907l_bool(cOMBB_1795223523l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc_010) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 676 (all Q_1 all P_1 ((all A_2 all B_4 (is_int(A_2) & is_int(B_4) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_4)) -> hBOOL(hAPP_rat_bool(P_1,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_4)))))) -> hBOOL(hAPP_rat_bool(P_1,Q_1)))) # label(fact_4728_Rat__induct__pos) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 677 (all N_13 all A_7 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_7)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_7),hAPP_nat_nat(suc,N_13)))))) # label(fact_3123_power__gt1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 678 (all D all B all C all A (zero_zero_nat != A -> (C != zero_zero_nat -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C),D)) -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,D),C) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B),A) <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B),C) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A),D))))))) # label(fact_2662_dvd__div__div__eq__mult) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 679 (all X_2 all Y_2 (is_int(X_2) & is_int(Y_2) -> (hAPP_int_real(number267125858f_real,X_2) = hAPP_int_real(number267125858f_real,Y_2) <-> Y_2 = X_2))) # label(fact_248_eq__number__of) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 680 (all Lx all Rx all Ry hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx),Ry)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx),Ry))) # label(fact_183_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 681 (all M_6 all N_20 hAPP_n1108039445umeral(semiri1619134803umeral,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_6),N_20)) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,hAPP_n1108039445umeral(semiri1619134803umeral,M_6)),N_20)) # label(fact_2980_of__nat__power) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 682 (all M all N_1 hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),hAPP_nat_nat(suc,N_1))) # label(fact_2870_add__Suc__right) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 683 (all A all E_2 all C all B all D (hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),E_2)),C) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),E_2)),D) <-> C = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,B),A)),E_2)),D))) # label(fact_1550_eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 684 (all A all B all C all D (hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),D) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C),D))))) # label(fact_2095_diff__eq__diff__less__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 685 (all P all Q all R hAPP_int_real(hAPP_f269654879t_real(hAPP_f766218899t_real(cOMBB_real_real_int,P),Q),R) = hAPP_real_real(P,hAPP_int_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__Int__Oin) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 686 (all M hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),hAPP_nat_nat(suc,zero_zero_nat)) = hAPP_nat_nat(suc,zero_zero_nat)) # label(fact_4342_gcd__Suc__0) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 687 (all A_3 all C_1 all D_3 all B_2 (is_int(B_2) & is_int(D_3) -> (B_2 != zero_zero_int -> (zero_zero_int != D_3 -> hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),D_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_1),B_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),D_3)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)),hAPP_int_rat(hAPP_int_fun_int_rat(fract,C_1),D_3)))))) # label(fact_4718_add__rat) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 688 (all L all U hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(ord_at641636577an_int(L),U)))) # label(fact_4915_finite__atLeastLessThan__int) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 689 (all X_30 all N_34 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_34)) | X_30 = one_one_real -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_30),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_30),N_34))))) # label(fact_992_dvd__power) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 690 (all N_16 all M_5 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_16),M_5)) -> hAPP_nat_complex(semiri2020571505omplex,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_5),N_16)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_nat_complex(semiri2020571505omplex,M_5)),hAPP_nat_complex(semiri2020571505omplex,N_16)))) # label(fact_3102_of__nat__diff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 691 (all D_3 all I (is_int(I) -> (I != zero_zero_int -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_3),I)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,D_3)),hAPP_int_int(abs_abs_int,I))))))) # label(fact_3433_dvd__imp__le__int) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 692 (all F all X_2 all L (hBOOL(hAPP_real_bool(deriv_real(F,X_2),L)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),L)) -> (exists D_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2)) & (all H_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H_1),D_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_2),H_1))),hAPP_real_real(F,X_2))))))))))) # label(fact_3886_DERIV__pos__inc__left) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 693 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A),B)),zero_zero_int))))) # label(fact_2556_neg__imp__zdiv__neg__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 694 (all A_3 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,A_3)),X_1)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,natfloor(X_1)),A_3) = natfloor(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),hAPP_nat_real(real_nat,A_3))))) # label(fact_3691_natfloor__subtract) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 695 (all A all B (is_int(B) & is_int(A) -> (B = A <-> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B) = zero_zero_int))) # label(fact_2086_eq__iff__diff__eq__0) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 696 (all X_2 all Y_2 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hAPP_real_real(hAPP_n546711566l_real(root,N),X_2) = hAPP_real_real(hAPP_n546711566l_real(root,N),Y_2) <-> Y_2 = X_2))) # label(fact_3897_real__root__eq__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 697 (all A (hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially)) -> (hBOOL(monoseq_real(A)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(A,zero_zero_nat)),zero_zero_real)) -> (all N_2 hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A))),hAPP_r1134773055l_bool(ord_at1589558736t_real(hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_2)),one_one_nat)))),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_2))))))))))) # label(fact_5061_summable__Leibniz_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 698 (all K_2 all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_1))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),N_1))))) # label(fact_1823_dvd__mult__cancel) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 699 (all A_176 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,zero_zero_int),A_176) = zero_zero_int) # label(fact_1043_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 700 (all K_1 all F all N (is_int(K_1) -> ((all I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),N)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_2),one_one_nat))),hAPP_nat_int(F,I_2)))),one_one_int)))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,zero_zero_nat)),K_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),hAPP_nat_int(F,N))) -> (exists I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),N)) & hAPP_nat_int(F,I_2) = K_1))))))) # label(fact_3478_int__val__lemma) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 701 (all A_102 all C_45 all B_85 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_102),C_45)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_85),C_45))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_102),B_85)))) # label(fact_2001_add__le__imp__le__right) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 702 (all N_1 all M -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,N_1)),hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,M))))) # label(fact_3530_not__int__zless__negative) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 703 (all M all K_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(fact_fact_int,M)),hAPP_int_int(fact_fact_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M),one_one_int)),K_2))))))) # label(fact_4285_fact__less__mono__int__aux) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 704 (all C_66 all D_13 all A_132 all B_111 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_132),B_111)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_66),D_13)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_111)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_66)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_132),C_66)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_111),D_13)))))))) # label(fact_1409_mult__strict__mono) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 705 (all N_1 -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_1))) # label(fact_1683_less__not__refl) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 706 (all N_63 all M_23 all X_42 all Y_34 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,X_42),Y_34)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_63),M_23)) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_42),N_63)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_34),M_23)))))) # label(fact_564_dvd__power__le) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 707 (all A_32 all B_28 (B_28 != zero_z126310315umeral -> hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_32),B_28)),B_28) = A_32)) # label(fact_2571_div__mult__self2__is__id) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 708 (all A_9 all N_15 all B_8 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_9),hAPP_nat_nat(suc,N_15))),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_8),hAPP_nat_nat(suc,N_15)))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_8)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_9),B_8))))) # label(fact_3108_power__le__imp__le__base) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 709 (all M all N_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N_1)),M))) # label(fact_4298_gcd__dvd1__nat) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 710 (all A all B all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma)) -> (hAPP_int_int(div_mod_int(A),Ma) = hAPP_int_int(div_mod_int(B),Ma) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma))))) # label(fact_2454_zcong__zmod__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 711 (all X_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_complex_real(re,X_1))),hAPP_complex_real(norm_norm_complex,X_1)))) # label(fact_4127_abs__Re__le__cmod) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 712 (all B_1_1 all B_2_1 is_bool(hAPP_P62333565t_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__prod_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__prod_Itc__) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 713 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,X_1)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sqrt,X_1)),X_1))) # label(fact_3849_sqrt__divide__self__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 714 (all V_14 all W_12 hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_14),W_12)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_14)),hAPP_int_int(number_number_of_int,W_12))) # label(fact_429_number__of__mult) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 715 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> X_1 = hAPP_real_real(arctan,hAPP_real_real(tan,X_1))))) # label(fact_3621_arctan__tan) # label(axiom) # label(non_clause). [assumption]. 5.34/5.41 716 (all F all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)) -> ((all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),B)) -> hBOOL(hAPP_real_bool(isCont_real_real(F),X)))) -> ((all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),X)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),B)) -> hBOOL(hAPP_real_bool(deriv_real(F,X),zero_zero_real)))) -> hAPP_real_real(F,A) = hAPP_real_real(F,B))))) # label(fact_5177_DERIV__isconst__end) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 717 (all P (is_bool(P) -> P = fTrue | fFalse = P)) # label(help_If_3_1_If_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_T) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 718 (all C all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),A)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B))) <-> zero_zero_real = C | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A),B)))) # label(fact_1374_dvd__mult__cancel__left) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 719 (all B_94 all C_52 all A_115 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_115)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_94),C_52)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_94),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_115),C_52)))))) # label(fact_1874_add__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 720 (all M one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),one_one_int)) # label(fact_4976_gcd__1__int) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 721 (all B_70 all A_81 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_81),zero_zero_rat)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_70),zero_zero_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_81),B_70)),zero_zero_rat))))) # label(fact_2189_add__neg__neg) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 722 (all P all Q P = hAPP_nat_rat(cOMBK_rat_nat(P),Q)) # label(help_COMBK_1_1_COMBK_000tc__Rat__Orat_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 723 (all A (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),A)),zero_zero_rat)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A),zero_zero_rat)))) # label(fact_2198_double__add__less__zero__iff__single__add__less__zero) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 724 (all N_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(fact_fact_nat,N_1)))) # label(fact_3810_fact__ge__Suc__0__nat) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 725 (all X_1 hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),pi)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(cos,X_1))) # label(fact_3569_cos__periodic__pi) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 726 (all X_3 (hBOOL(summable_complex(X_3)) -> hAPP_complex_real(im,hAPP_f213450978omplex(suminf_complex,X_3)) = hAPP_f352196356l_real(suminf_real,hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,im),X_3)))) # label(fact_4848_Im_Osuminf) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 727 (all N all Ma (hBOOL(hAPP_int_bool(nat_is_nat,Ma)) -> (hBOOL(hAPP_int_bool(nat_is_nat,N)) -> image_nat_int(semiri1621563631at_int,hAPP_n1699378549t_bool(ord_at238088361st_nat(hAPP_int_nat(nat,Ma)),hAPP_int_nat(nat,N))) = hAPP_i1948725293t_bool(ord_at875362053st_int(Ma),N)))) # label(fact_5127_SetInterval_Otransfer__int__nat__set__functions) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 728 (all N all P_1 (-hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(P_1,N)) -> (exists K (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K),N)) & (all I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),K)) -> -hBOOL(hAPP_nat_bool(P_1,I_2)))) & hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)))))))) # label(fact_1827_ex__least__nat__less) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 729 (all K_2 all L_2 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(bit0,K_2)),hAPP_int_int(bit1,L_2)) = hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(pred,K_2)),L_2))) # label(fact_4048_diff__bin__simps_I8_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 730 (all A_186 all K_8 (is_int(A_186) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_186),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),K_8))),zero_zero_int)) -> zero_zero_int = A_186))) # label(fact_899_even__power__le__0__imp__0) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 731 (all X_2 all P_5 (is_int(X_2) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),hAPP_i1948725293t_bool(sr,P_5))) -> hAPP_int_int(standardRes(P_5),X_2) = X_2))) # label(fact_2297_StandardRes__SR__prop) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 732 (all N_1 all M hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,M),hAPP_int_int(div_mod_int(M),N_1)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N_1),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,M),N_1))) # label(fact_2642_zmult__div__cancel) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 733 (all C_41 all A_98 all B_81 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_41),A_98)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_41),B_81))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_98),B_81)))) # label(fact_2039_add__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 734 (all D_3 all A_3 all B_2 (one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2)) -> one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),B_2))) # label(fact_4351_coprime__rmult__nat) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 735 (all X_2 all Y_2 all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B),X_2)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B),Y_2))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),Y_2))))) # label(fact_878_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 736 (all B_114 all A_137 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_137)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_114),zero_zero_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_137),B_114)),zero_zero_rat))))) # label(fact_1313_mult__pos__neg) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 737 (all N_12 all R_6 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),R_6)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,R_6),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,R_6),hAPP_nat_nat(suc,N_12))),R_6))))) # label(fact_3159_realpow__Suc__le__self) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 738 (all X_3 all A (hBOOL(tendsto_nat_complex(X_3,A,sequentially)) -> hBOOL(tendsto_nat_complex(hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,cnj),X_3),hAPP_complex_complex(cnj,A),sequentially)))) # label(fact_5054_cnj_OLIMSEQ) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 739 (all C_34 all D_6 all A_79 all B_68 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_79),B_68)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_34),D_6)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_79),C_34)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_68),D_6)))))) # label(fact_2207_add__le__less__mono) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 740 (all Q_2 all R_3 hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,Q_2),R_3) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Q_2),hAPP_rat_rat(inverse_inverse_rat,R_3))) # label(fact_4650_divide__rat__def) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 741 (all A_3 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(inv(P_3),A_3)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))))))) # label(fact_2132_inv__less__p__minus__1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 742 (all M all N_1 all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),N_1)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),M))))) # label(fact_4170_prime__dvd__mult) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 743 (all C_65 all D_12 all A_131 all B_110 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_131),B_110)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_65),D_12)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_131)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_65)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_131),C_65)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_110),D_12)))))))) # label(fact_1418_mult__strict__mono_H) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 744 (all M all N_1 all K_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2)) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),hAPP_int_int(legacy_zgcd(M),N_1)) = hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N_1)))) # label(fact_4509_zgcd__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 745 (all B_92 all A_113 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_113)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_92)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_113),B_92)))))) # label(fact_1890_add__pos__nonneg) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 746 (all N_1 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),X_1)),one_one_real)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),one_one_real)),N_1))))) # label(fact_3711_LIMSEQ__inverse__realpow__zero__lemma) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 747 (all M_12 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(semiri132038758t_real,M_12)))) # label(fact_2915_zero__le__imp__of__nat) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 748 (all Ma all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Ma)))) # label(fact_1798_add__gr__0) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 749 (all X_1 all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_1)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),Y_1)))) # label(fact_3361_real__gt__half__sum) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 750 (all A all C all B (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B),C))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A),B)))) # label(fact_2077_add__less__cancel__right) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 751 (all P all Q all R hAPP_f1690286043t_real(P,hAPP_i371647666l_real(Q,R)) = hAPP_i447943416t_real(hAPP_f1295083291t_real(hAPP_f212164053t_real(cOMBB_929094628al_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc__032) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 752 (all X_1 all Y_1 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sqrt,X_1)),hAPP_real_real(sqrt,Y_1)) = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),Y_1))) # label(fact_3318_real__sqrt__mult) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 753 (all Ta all A_1 all D_1 (is_int(Ta) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ta),one_one_int)),A_1)) -> (all X (is_int(X) -> ((all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A_1)) -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) != X)))))) -> (Ta = X -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),D_1) = Ta)))))))) # label(fact_5093_aset_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 754 (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),X_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),one_one_real)) -> hBOOL(hAPP_real_bool(deriv_real(arccos,X_2),hAPP_real_real(inverse_inverse_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))))))) # label(fact_3860_DERIV__arccos) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 755 (all X_2 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (X_2 = one_one_real <-> hAPP_real_real(hAPP_n546711566l_real(root,N),X_2) = one_one_real))) # label(fact_3909_real__root__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 756 (all N_1 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> hAPP_real_real(ln,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),hAPP_real_real(ln,X_1)))) # label(fact_3696_ln__realpow) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 757 (all A_71 all B_63 all C_31 hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_71),B_63)),C_31) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_71),C_31)),B_63)),C_31)) # label(fact_2320_mod__mult__left__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 758 (all X_1 hAPP_complex_complex(uminus473333897omplex,X_1) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),X_1))) # label(fact_3977_complex__i__mult__minus) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 759 (all X_11 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_11)) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat,X_11))),hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat,X_11))) = hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_11))))) # label(fact_2810_of__nat__double) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 760 (all C all A all B (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C),A)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C),B))) <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A),B)))) # label(fact_2070_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 761 (all A_121 all C_56 all B_101 all D_9 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_121),B_101)),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,C_56),D_9)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_121),C_56)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_101),D_9))) # label(fact_1526_add__diff__add) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 762 (all M all N_1 hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(suc,M)),N_1)) # label(fact_2869_add__Suc) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 763 (all K_2 all L_2 (is_int(L_2) -> (zero_zero_int != L_2 -> posDivAlg(hAPP_int_int(abs_abs_int,K_2),hAPP_int_int(abs_abs_int,L_2)) = pdivmod(K_2,L_2)) & (L_2 = zero_zero_int -> pdivmod(K_2,L_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),hAPP_int_int(abs_abs_int,K_2))))) # label(fact_3943_pdivmod__posDivAlg) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 764 (all C_57 all A_123 all B_102 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_57),A_123)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_57),B_102))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_57)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_123),B_102))))) # label(fact_1463_mult__left__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 765 (all J_2 all A_3 all K_2 all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(K_2),zero_zero_int),P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,J_2),K_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),K_2))),K_2)),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,J_2),K_2)),A_3),P_3))))))) # label(fact_2221_aux______2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 766 (all I_1 all P_1 all K_1 (hBOOL(hAPP_nat_bool(P_1,K_1)) -> ((all N_2 (hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,N_2))) -> hBOOL(hAPP_nat_bool(P_1,N_2)))) -> hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,K_1),I_1)))))) # label(fact_3268_zero__induct__lemma) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 767 (all Ma all Z_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_int_nat(nat,Z_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,Ma)),Z_2)))) # label(fact_3127_zless__nat__eq__int__zless) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 768 (all A_67 all C_27 all B_59 hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_67),B_59)),C_27) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(div_mod_nat(A_67),C_27)),B_59)),C_27)) # label(fact_2335_zmod__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 769 (all B_1_1 all B_2_1 is_bool(hAPP_P880235623l_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__prod_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__fun_Itc__I) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 770 (all X_1 all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> (exists N_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),Y_1)),X_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_2))),Y_1)))))))) # label(fact_3748_reals__Archimedean4) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 771 (all B_92 all A_113 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_113)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_92)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_113),B_92)))))) # label(fact_1891_add__pos__nonneg) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 772 (all Z1 all Z2 all Z3 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z2),Z3)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),Z2)),Z3)) # label(fact_263_zadd__assoc) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 773 (all B_127 all A_151 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_151)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_127),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_127),A_151)),zero_zero_real))))) # label(fact_1229_mult__nonneg__nonpos2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 774 (all K_1 all Ma all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K_1)),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K_1)),N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)))) # label(fact_3067_Suc__mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 775 (all B all D all A (D = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A),B) <-> (all E (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,E),B)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,E),A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,E),D)))) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),B)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),A)))) # label(fact_4325_gcd__unique__nat) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 776 (all V_5 all W_3 all C_54 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_5),W_3))),C_54) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_5)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(number_number_of_int,W_3)),C_54))) # label(fact_1581_add__number__of__diff1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 777 (all A all B all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X_2),A)),hAPP_real_real(hAPP_r1250527377l_real(powr,X_2),B)))))) # label(fact_4410_powr__less__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 778 (all Ma all N (Ma != N <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),Ma)))) # label(fact_1684_nat__neq__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 779 (all M all N_1 all K_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),N_1)),hAPP_nat_nat(suc,K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)),K_2)) # label(fact_2880_Suc__diff__diff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 780 (all A all X_2 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zOdd)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(uminus_uminus_int,A)),hAPP_int_nat(nat,X_2)) = hAPP_int_int(uminus_uminus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),hAPP_int_nat(nat,X_2)))))) # label(fact_3554_neg__odd__power) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 781 (all Y_2 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),hAPP_int_rat(number_number_of_rat,Y_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,pls)),Y_2)))) # label(fact_289_less__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 782 (all X_2 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(exp_real,X_2)),hAPP_real_real(exp_real,Y_2))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Y_2)))) # label(fact_3824_exp__less__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 783 (all N_26 hAPP_int_nat(number_number_of_nat,hAPP_nat_int(semiri1621563631at_int,N_26)) = hAPP_nat_nat(semiri984289939at_nat,N_26)) # label(fact_2824_number__of__int) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 784 (all M_7 all N_21 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(semiri2020571505omplex,M_7)),hAPP_nat_complex(semiri2020571505omplex,N_21)) = hAPP_nat_complex(semiri2020571505omplex,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_7),N_21))) # label(fact_2965_of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 785 (all A_176 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,zero_zero_rat),A_176) = zero_zero_rat) # label(fact_1049_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 786 (all M all N_1 hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M),N_1)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_nat_real(real_nat,M)),N_1)) # label(fact_3653_power__real__of__nat) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 787 (all B_119 all A_142 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_142),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_119),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_142),B_119)))))) # label(fact_1279_mult__neg__neg) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 788 (all W hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),W))) # label(fact_1523_real__le__refl) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 789 (all X_1 (-hBOOL(hAPP_int_bool(even_odd_even_int,X_1)) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),one_one_int))) # label(fact_3243_odd__plus__one__div__two) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 790 (all X_1 hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,X_1))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(im,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_complex_real(im,hAPP_complex_complex(invers1449016382omplex,X_1))) # label(fact_4111_complex__Im__inverse) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 791 (all B_72 all C_35 all A_83 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_83)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_72),C_35)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_72),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_83),C_35)))))) # label(fact_2162_add__increasing) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 792 (all A hBOOL(tendst1507391555omplex(cnj,hAPP_complex_complex(cnj,A),at_complex(A)))) # label(fact_5077_cnj_Ocont) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 793 (all C_72 all B_124 all A_147 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_124),A_147)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_72),zero_zero_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_72),A_147)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_72),B_124)))))) # label(fact_1256_mult__strict__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 794 (all X_2 hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_2)) = hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,uminus_uminus_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),sin_coeff)),hAPP_r474017924t_real(power_power_real,X_2))))) # label(fact_4822_lemma__sin__minus) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 795 (all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),M)) -> hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,M),one_one_int)) = hAPP_nat_nat(suc,hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,M),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_3246_inv__is__inv__aux) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 796 (all M hAPP_nat_nat(suc,M) != zero_zero_nat) # label(fact_2855_Zero__not__Suc) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 797 (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,X_2)),one_one_real)))) # label(fact_3840_exp__le__one__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 798 (all Ma all X_2 ((exists Y (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X_2),Ma)) & is_int(Y))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,Ma),X_2)))) # label(fact_1627_QuadRes__def) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 799 (all Z_2 (is_int(Z_2) -> (one_one_int = hAPP_int_int(abs_abs_int,Z_2) <-> Z_2 = hAPP_int_int(number_number_of_int,min) | one_one_int = Z_2))) # label(fact_3436_abs__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 800 (all X_48 all N_69 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_48),N_69)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_48),N_69)) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_48),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_69))) # label(fact_43_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 801 (all N all A (hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially)) -> ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(A,N_2)))) -> ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A,hAPP_nat_nat(suc,N_2))),hAPP_nat_real(A,N_2)))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N)))),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)))))))) # label(fact_5059_summable__Leibniz_H_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 802 (all N_63 all M_23 all X_42 all Y_34 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_42),Y_34)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_63),M_23)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_42),N_63)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_34),M_23)))))) # label(fact_566_dvd__power__le) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 803 (all N_54 all A_204 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_204)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_204),N_54))))) # label(fact_665_zero__less__power) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 804 (all A_219 hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_219),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_219),A_219)),A_219)) # label(fact_479_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 805 (all Z_12 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Z_12),Z_12) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_12)) # label(fact_491_semiring__mult__2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 806 (all M hAPP_nat_nat(div_mod_nat(M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,hAPP_nat_nat(suc,M))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_3204_mod2__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 807 (all Ma all X_3 (hBOOL(hAPP_f448129468l_bool(resSet(Ma),X_3)) <-> (all Y1 all Y2 (is_int(Y1) & is_int(Y2) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y2),X_3)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Y1),Y2),Ma)) & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y1),X_3)) -> Y2 = Y1))))) # label(fact_4053_ResSet__def) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 808 (all A_3 all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),P_3)))))) # label(fact_2776_Euler__part2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 809 (all P all Q all R hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,P),Q),R) = hAPP_bool_bool(hAPP_n1006566506l_bool(P,R),hAPP_nat_bool(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__HOL__Obool_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 810 (all B_1_1 all B_2_1 is_bool(hAPP_rat_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__Rat__Orat_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 811 (all X_2 all Y_2 (X_2 = Y_2 <-> hAPP_real_real(exp_real,X_2) = hAPP_real_real(exp_real,Y_2))) # label(fact_3826_exp__inj__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 812 (all X_2 -(hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven)) & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zOdd)))) # label(fact_2905_even__odd__conj) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 813 (all X_2 all Y_2 (is_int(Y_2) & is_int(X_2) -> (zero_zero_int != X_2 | zero_zero_int != Y_2 <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))))) # label(fact_854_sum__power2__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 814 (all K_2 all I all J_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),J_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),I)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),J_2)))))) # label(fact_718_zmult__zless__mono2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 815 (all X_11 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_11)) -> hAPP_n1108039445umeral(semiri1619134803umeral,hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_11))) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_n1108039445umeral(semiri1619134803umeral,hAPP_int_nat(nat,X_11))),hAPP_n1108039445umeral(semiri1619134803umeral,hAPP_int_nat(nat,X_11))))) # label(fact_2809_of__nat__double) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 816 (all A_10 all N_17 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_10),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_10),N_17)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_10),hAPP_nat_nat(suc,N_17))) # label(fact_3037_power__Suc) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 817 (all P all Q all R hAPP_int_bool(hAPP_f2119767738t_bool(hAPP_f423804115t_bool(cOMBB_1418110531ol_int,P),Q),R) = hAPP_f448129468l_bool(P,hAPP_i1948725293t_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__HOL__Obo) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 818 (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),X_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),one_one_real)) -> hBOOL(hAPP_real_bool(isCont_real_real(arcsin),X_2))))) # label(fact_5172_isCont__arcsin) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 819 (all X_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),natceiling(X_1)))) # label(fact_3389_zero__le__natceiling) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 820 (all X_3 (hBOOL(summable_complex(X_3)) -> hBOOL(summable_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,im),X_3))))) # label(fact_4846_Im_Osummable) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 821 (all N ((exists M_1 N = hAPP_nat_nat(suc,M_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)))) # label(fact_3048_gr0__conv__Suc) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 822 (all X_2 all Y_2 all Z_2 (X_2 = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,Y_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_2)) <-> hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,Y_2),Z_2) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_2))) # label(fact_3367_eq__divide__2__times__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 823 (all X_1 all Y_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3346_real__sqrt__sum__squares__ge1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 824 (all X_1 all Y_1 hAPP_complex_real(im,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(re,X_1)),hAPP_complex_real(im,Y_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(im,X_1)),hAPP_complex_real(re,Y_1)))) # label(fact_4129_complex__Im__mult) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 825 (all B_2 hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),zero_zero_int) = posDivAlg(zero_zero_int,B_2)) # label(fact_2520_posDivAlg__0) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 826 (all X_1 (hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X_1),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))) = X_1)) # label(fact_3779_even__nat__div__two__times__two) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 827 (all A all C all D all B (is_int(B) & is_int(D) -> (zero_zero_int != B -> (zero_zero_int != D -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),D)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),D))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),D)))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B)),hAPP_int_rat(hAPP_int_fun_int_rat(fract,C),D)))))))) # label(fact_4717_le__rat) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 828 (all W (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(number_number_of_int,W))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(number_number_of_int,W))) -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int) = posDivAlg(one_one_int,hAPP_int_int(number_number_of_int,W))) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(number_number_of_int,W))) -> posDivAlg(one_one_int,hAPP_int_int(number_number_of_int,W)) = hAPP_P1975530577nt_int(adjust(hAPP_int_int(number_number_of_int,W)),posDivAlg(one_one_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(number_number_of_int,W))))))) # label(fact_2501_posDivAlg__eqn__1__number__of) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 829 (all C_78 all D_14 all A_159 all B_135 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_159),B_135)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_78),D_14)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_159)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_78)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_159),C_78)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_135),D_14)))))))) # label(fact_1192_mult__mono_H) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 830 (all N all V hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(suc,N)),hAPP_int_nat(number_number_of_nat,V)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))),zero_zero_nat),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,N),hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))))) # label(fact_5118_min__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 831 (all Ma all N ((exists J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),N)) & hAPP_nat_nat(suc,J_1) = Ma)) | zero_zero_nat = Ma <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_nat_nat(suc,N))))) # label(fact_3050_less__Suc__eq__0__disj) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 832 (all F all G ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,N_2)),hAPP_nat_real(G,N_2)))) -> (hBOOL(summable_real(F)) -> (hBOOL(summable_real(G)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f352196356l_real(suminf_real,F)),hAPP_f352196356l_real(suminf_real,G))))))) # label(fact_4880_summable__le) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 833 (all A all E_2 all C all B all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),E_2)),C)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),E_2)),D))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B),A)),E_2)),D))))) # label(fact_1569_le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 834 (all A_29 all B_25 all C_10 (C_10 != zero_zero_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_29),B_25) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_10),A_29)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_10),B_25)))) # label(fact_2582_div__mult__mult1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 835 (all K_1 all Ma all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)))) # label(fact_1764_nat__add__left__cancel__less) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 836 (all M all N_1 all K_2 hAPP_int_int(legacy_zgcd(M),N_1) = hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N_1),K_2))),N_1)) # label(fact_4499_zgcd__zadd__zmult) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 837 (all N_58 ((N_58 != zero_zero_nat -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,zero_zero_int),N_58) = zero_zero_int) & (N_58 = zero_zero_nat -> one_one_int = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,zero_zero_int),N_58)))) # label(fact_594_power__0__left) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 838 (all K_1 all N all F_2 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F_2),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),K_1)))),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F_2),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),K_1))) = hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,F_2),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),K_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))) # label(fact_4897_sumr__offset2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 839 (all A_58 all C_21 all B_50 hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_58),C_21)),B_50)),C_21) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_58),B_50)),C_21)) # label(fact_2368_zmod__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 840 (all P all Q all R hAPP_int_int(P,hAPP_int_int(Q,R)) = hAPP_int_int(hAPP_f1356864277nt_int(hAPP_f747162069nt_int(cOMBB_int_int_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__Int__Oint_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 841 (all X_1 all Y_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_1),Y_1)) -> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_1) = hAPP_int_int(abs_abs_int,X_1))) # label(fact_4994_gcd__proj1__if__dvd__int) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 842 (all N_1 hAPP_nat_nat(suc,N_1) != N_1) # label(fact_2839_n__not__Suc__n) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 843 (all Diff all F all N all H ((all M_1 all T (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),H)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),T)) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),T),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T))))) & F = hAPP_n546711566l_real(Diff,zero_zero_nat) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H)) -> (exists T (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),T)) & hAPP_real_real(F,H) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,H))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),N))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,T),H)))))) # label(fact_4933_Maclaurin__objl) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 844 (all M all N_1 ((-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) -> hAPP_nat_nat(div_mod_nat(M),N_1) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)),N_1)) & (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) -> M = hAPP_nat_nat(div_mod_nat(M),N_1)))) # label(fact_2515_mod__if) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 845 (all Ma all D ((exists Q_4 Ma = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,D),Q_4)) <-> hAPP_nat_nat(div_mod_nat(Ma),D) = zero_zero_nat)) # label(fact_2513_mod__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 846 (all C_2 hBOOL(hAPP_f448129468l_bool(nat_nat_set,image_nat_int(semiri1621563631at_int,C_2)))) # label(fact_4614_Nat__Transfer_Otransfer__int__nat__set__function__closures_I5_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 847 (all A_193 all M_21 all N_45 hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_193),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_21),N_45)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_193),M_21)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_193),N_45))) # label(fact_818_power__add) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 848 (all Ma all N all A (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A)) -> (hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A),Ma) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A),N) <-> N = Ma))) # label(fact_877_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 849 (all N_1 hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,N_1))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),N_1)) # label(fact_3174_Suc3__eq__add__3) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 850 (all W_17 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,W_17)),hAPP_int_complex(number528085621omplex,W_17)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_int_complex(number528085621omplex,W_17)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_20_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 851 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X_1)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X_1))),X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3413_abs__ln__one__plus__x__minus__x__bound) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 852 (all M all N_1 all K_2 all L_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_2),L_2)) -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),L_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_2),N_1) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))))) # label(fact_1759_less__add__eq__less) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 853 (all P all Q all R hAPP_f766218899t_real(P,hAPP_i371647666l_real(Q,R)) = hAPP_i1671359920t_real(hAPP_f74147283t_real(hAPP_f1169087445t_real(cOMBB_2129825580al_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc__031) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 854 (all M_12 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(semiri984289939at_nat,M_12)))) # label(fact_2916_zero__le__imp__of__nat) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 855 (all A_87 all B_76 A_87 = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_87),B_76)),B_76)) # label(fact_2136_add__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 856 (all U all Ma all N all I_1 all J (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J)) -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),I_1)),U)),N) = Ma <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U)),N)))) # label(fact_1847_nat__eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 857 (all A_175 all E_3 all B_149 all C_87 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_175),E_3)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_149),E_3)),C_87)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_175),B_149)),E_3)),C_87)) # label(fact_1068_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 858 (all A_47 hAPP_int_int(div_mod_int(A_47),one_one_int) = zero_zero_int) # label(fact_2415_mod__by__1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 859 (all A_189 all N_40 all N_39 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_40),N_39)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),A_189)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_189),N_40)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_189),N_39)))))) # label(fact_866_power__increasing) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 860 (all Z_10 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_10),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_10),Z_10)) # label(fact_496_semiring__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 861 (all B_150 all A_178 (zero_zero_real != A_178 -> (zero_zero_real != B_150 -> zero_zero_real != hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_178),B_150)))) # label(fact_1027_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 862 (all A all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),zero_zero_int),Ma)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),A)))) # label(fact_1398_zcong__zero__equiv__div) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 863 (all Y_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arctan,Y_1)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_3470_arctan__ubound) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 864 (all K_1 all N all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),Ma)))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),N)))) # label(fact_667_zdvd__reduce) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 865 (all X_2 (zero_zero_rat = X_2 <-> zero_zero_rat = X_2)) # label(fact_1902_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 866 (all X_2 hAPP_real_real(cos,X_2) = hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),cos_coeff)),hAPP_r474017924t_real(power_power_real,X_2)))) # label(fact_4820_cos__def) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 867 (all N_1 all K_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),N_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(legacy_zgcd(M),N_1)))))) # label(fact_4489_zgcd__greatest) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 868 (all C_79 all D_15 all A_160 all B_136 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_160),B_136)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_79),D_15)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_136)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_79)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_160),C_79)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_136),D_15)))))))) # label(fact_1182_mult__mono) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 869 (all X_2 all A_1 (hBOOL(hAPP_nat_bool(A_1,X_2)) <-> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A_1)))) # label(fact_202_mem__def) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 870 (all A_74 (is_int(A_74) -> A_74 = hAPP_int_int(div_mod_int(A_74),zero_zero_int))) # label(fact_2306_mod__by__0) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 871 (all F ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(F,N_2)))) -> (hBOOL(summable_real(F)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_f352196356l_real(suminf_real,F)))))) # label(fact_4859_suminf__0__le) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 872 (all X_2 (hAPP_real_real(cos,X_2) = zero_zero_real <-> (exists N_2 (hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = X_2 & -hBOOL(hAPP_nat_bool(even_odd_even_nat,N_2)))) | (exists N_2 (-hBOOL(hAPP_nat_bool(even_odd_even_nat,N_2)) & hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = X_2)))) # label(fact_3753_cos__zero__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 873 (all A all X_3 all X_2 (hBOOL(tendsto_nat_real(X_3,X_2,sequentially)) -> ((exists N_3 all N_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_3),N_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),hAPP_nat_real(X_3,N_2))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X_2))))) # label(fact_5033_LIMSEQ__le__const) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 874 (all M_8 all N_22 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(semiri984289939at_nat,M_8)),hAPP_nat_nat(semiri984289939at_nat,N_22)) = hAPP_nat_nat(semiri984289939at_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_8),N_22))) # label(fact_2960_of__nat__add) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 875 (all X_2 (is_int(X_2) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven)) -> -(all K (is_int(K) -> X_2 != hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),K)))))) # label(fact_3247_zEvenE) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 876 (all A_32 all B_28 (is_int(B_28) & is_int(A_32) -> (B_28 != zero_zero_int -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_32),B_28)),B_28) = A_32))) # label(fact_2569_div__mult__self2__is__id) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 877 (all A_3 all B_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,B_2),A_3)) # label(fact_4304_gcd__commute__nat) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 878 (all A_3 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,A_3)),X_1)) -> natceiling(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),hAPP_nat_real(real_nat,A_3))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,natceiling(X_1)),A_3))) # label(fact_3693_natceiling__subtract) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 879 (all A_21 all C_6 all B_18 (B_18 != zero_zero_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_21),B_18)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_21),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_6),B_18))),B_18))) # label(fact_2648_div__mult__self1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 880 (all C_41 all A_98 all B_81 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_41),A_98)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_41),B_81))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_98),B_81)))) # label(fact_2038_add__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 881 (all A_127 all C_61 all B_106 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_127),C_61)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_106),C_61))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_61)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_127),B_106))))) # label(fact_1438_mult__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 882 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X_1)),hAPP_real_real(exp_real,X_1))))) # label(fact_3852_exp__ge__add__one__self__aux) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 883 (all N_1 all F_3 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,F_3),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,F_3),N_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,F_3),hAPP_int_int(div_mod_int(M),N_1)))))) # label(fact_2404_zdvd__zmod) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 884 (all A_124 all C_58 all B_103 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_124),C_58)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_103),C_58))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_58)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_124),B_103))))) # label(fact_1455_mult__right__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 885 (all R_3 all A_3 hAPP_real_real(abs_abs_real,R_3) = hAPP_complex_real(norm_norm_complex,rcis(R_3,A_3))) # label(fact_4258_complex__mod__rcis) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 886 (all C all D all A all B (C != D & A != B <-> hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A),D)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B),C)) != hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A),C)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B),D)))) # label(fact_318_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 887 (all A_107 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_107),zero_zero_rat) = A_107) # label(fact_1969_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 888 (all A all N (N != zero_zero_nat & zero_zero_complex = A <-> zero_zero_complex = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A),N))) # label(fact_550_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 889 (all A_3 all B_2 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),B_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),A_3)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) # label(fact_8_zadd__power3) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 890 (all A_16 all B_13 A_16 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_16),B_13)),B_13)),hAPP_nat_nat(div_mod_nat(A_16),B_13))) # label(fact_2676_mod__div__equality) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 891 (all A_3 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(div_mod_int(A_3),B_2)),B_2)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(div_mod_int(A_3),B_2))))) # label(fact_2452_pos__mod__conj) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 892 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3)) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),A_3) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),A_3)))) # label(fact_2534_pos__zdiv__mult__2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 893 (all X_1 all Y_1 (zero_zero_nat != Y_1 -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Y_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y_1),hAPP_nat_nat(div_mod_nat(X_1),Y_1)))) # label(fact_4360_gcd__non__0__nat) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 894 (all N_28 all K_5 all M_15 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,K_5),M_15)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,K_5),N_28)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,K_5),hAPP_C498520661umeral(div_mo1740067990umeral(M_15),N_28)))))) # label(fact_2382_dvd__mod) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 895 (all N_1 all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,P_3),N_1)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),N_1)))) # label(fact_4177_prime__coprime__strong) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 896 (all N_26 hAPP_int_real(number267125858f_real,hAPP_nat_int(semiri1621563631at_int,N_26)) = hAPP_nat_real(semiri132038758t_real,N_26)) # label(fact_2826_number__of__int) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 897 (all M all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),hAPP_nat_nat(suc,N_1))) # label(fact_3069_mult__Suc__right) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 898 (all W hAPP_complex_complex(cnj,hAPP_int_complex(number528085621omplex,W)) = hAPP_int_complex(number528085621omplex,W)) # label(fact_4066_complex__cnj__number__of) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 899 (all N_67 all A_218 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_218)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_218),N_67)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_218),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_218),N_67)))))) # label(fact_516_power__less__power__Suc) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 900 (all C_63 all D_10 all A_129 all B_108 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_129),B_108)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_63),D_10)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_129)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_63)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_129),C_63)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_108),D_10)))))))) # label(fact_1430_mult__le__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 901 (all Ma all N (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(semiri1619134803umeral,Ma)),hAPP_n1108039445umeral(semiri1619134803umeral,N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)))) # label(fact_2949_of__nat__le__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 902 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K1)),hAPP_int_int(bit1,K2))))) # label(fact_272_less__eq__int__code_I14_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 903 (all P all Q all R hAPP_f892584630t_bool(hAPP_P1400155668t_bool(P,R),hAPP_P1668131738nt_int(Q,R)) = hAPP_P989584703t_bool(hAPP_f1187443546t_bool(hAPP_f11709033t_bool(cOMBS_1676841879t_bool,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc_) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 904 (all M all N_1 -(all Q_4 all R_1 -hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Q_4),R_1))))) # label(fact_4220_divmod__nat__rel__ex) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 905 (all X_1 all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X_1)),natceiling(Y_1))))) # label(fact_3390_natceiling__mono) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 906 (all K_2 (is_int(K_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_2)) -> (exists N_2 (K_2 = hAPP_nat_int(semiri1621563631at_int,N_2) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2))))))) # label(fact_3250_zero__less__imp__eq__int) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 907 (all Y_1 all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X_1)) -> (exists N_2 (is_int(N_2) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Y_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N_2),X_1))))),X_1)))))) # label(fact_3475_best__division__abs) # label(axiom) # label(non_clause). [assumption]. 5.34/5.42 908 (all P_2 all P_1 ((all X (is_int(X) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(P_2,X)) <-> hBOOL(hAPP_int_bool(P_1,X)))))) -> collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),P_2)) = collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),P_1)))) # label(fact_4957_transfer__nat__int__set__cong) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 909 (all B_96 all A_117 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_117),zero_zero_rat)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_96),zero_zero_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_117),B_96)),zero_zero_rat))))) # label(fact_1867_add__neg__nonpos) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 910 (all Z_9 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Z_9),Z_9) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Z_9),hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_504_mult__2__right) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 911 (all X_11 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_11)) -> hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(semiri151668891at_rat,hAPP_int_nat(nat,X_11))),hAPP_nat_rat(semiri151668891at_rat,hAPP_int_nat(nat,X_11))) = hAPP_nat_rat(semiri151668891at_rat,hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_11))))) # label(fact_2815_of__nat__double) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 912 (all B_128 all A_152 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_152),zero_zero_rat)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_128)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_152),B_128)),zero_zero_rat))))) # label(fact_1226_mult__nonpos__nonneg) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 913 (all A (A = zero_zero_real <-> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),A) = zero_zero_real)) # label(fact_642_double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 914 (all A_3 all B_2 all Q_2 all R_3 (is_int(R_3) & is_int(B_2) -> (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3))) -> (B_2 != zero_zero_int -> hAPP_int_int(div_mod_int(A_3),B_2) = R_3)))) # label(fact_2795_divmod__int__rel__mod) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 915 (all A_191 all N_42 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_191),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_42))))) # label(fact_857_zero__le__even__power_H) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 916 (all A all B (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B)),zero_zero_rat)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A),B)))) # label(fact_2219_less__iff__diff__less__0) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 917 (all Lx_6 all Ly_4 all Rx_6 all Ry_4 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_6),Rx_6)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ly_4),Ry_4)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_6),Ly_4)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_6),Ry_4))) # label(fact_137_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 918 (all C_88 all A_180 all B_152 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_180),B_152)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,B_152),C_88)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_180),C_88))))) # label(fact_988_dvd__trans) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 919 (all A_3 all X_1 (is_int(X_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),A_3)) -> (X_1 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),one_one_int) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),one_one_int)))))))) # label(fact_1585_Euler_Oaux1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 920 (all Z_11 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Z_11),Z_11) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_11)) # label(fact_495_mult__2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 921 (all A_220 A_220 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_220),hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,pls)))) # label(fact_457_mult__numeral__1__right) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 922 (all P_3 all A_3 hAPP_int_int(div_mod_int(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),P_3) = hAPP_int_int(inv(P_3),A_3)) # label(fact_2774_WilsonRuss_Oinv__def) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 923 (all A all B all N (N != zero_zero_nat -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),N)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B),N))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),B))))) # label(fact_2269_pow__divides__eq__int) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 924 (all Y_1 Y_1 = hAPP_real_real(tan,hAPP_real_real(arctan,Y_1))) # label(fact_3597_tan__arctan) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 925 (all C_1 all A_3 all B_2 (A_3 = B_2 -> (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C_1),B_2)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),C_1)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C_1),A_3)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),C_1))))) # label(fact_1705_dvd_Oord__eq__less__trans) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 926 (all A_177 zero_zero_rat = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_177),zero_zero_rat)) # label(fact_1042_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 927 (all Y_6 all P_6 all N_10 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_6),N_10)) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),P_6))),Y_6) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N_10)),P_6)))) # label(fact_3181_lemma__realpow__diff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 928 (all X_1 (hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1)) -> hAPP_nat_nat(div_mod_nat(X_1),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) = zero_zero_nat)) # label(fact_3773_even__nat__mod__two__eq__zero) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 929 (all B_127 all A_151 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_151)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_127),zero_z891286103de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_127),A_151)),zero_z891286103de_int))))) # label(fact_1231_mult__nonneg__nonpos2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 930 (all M_22 all A_214 all N_62 all B_159 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_214),N_62)),B_159)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_22),N_62)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_214),M_22)),B_159))))) # label(fact_570_power__le__dvd) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 931 (all X_1 all Y_1 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) -> -(hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))))) # label(fact_1708_dvd_Oless__not__sym) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 932 (all A_3 all B_2 all C_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1)) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),C_1) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),C_1)))) # label(fact_2636_zdiv__zmult2__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 933 (all N all N_7 all F ((all N_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N_2)),hAPP_nat_nat(F,hAPP_nat_nat(suc,N_2)))) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,hAPP_nat_nat(suc,N_2))),hAPP_nat_nat(F,N_2))))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N_7)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N)),hAPP_nat_nat(F,N_7))) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N_7)),hAPP_nat_nat(F,N)))))) # label(fact_3262_dvd_Olift__Suc__mono__less) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 934 (all A_3 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2))) -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)) = negDivAlg(A_3,B_2)) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2))) -> hAPP_P1975530577nt_int(adjust(B_2),negDivAlg(A_3,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_2))) = negDivAlg(A_3,B_2)))) # label(fact_2496_negDivAlg__eqn) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 935 (all A_34 A_34 = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_34),one_on1684967323de_int)) # label(fact_2548_div__by__1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 936 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1))),one_one_real))))) # label(fact_3586_cos__double__less__one) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 937 (all X_43 all Y_35 all Q_10 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_43),Y_35)),Q_10) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_43),Q_10)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_35),Q_10))) # label(fact_346_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 938 (all A_23 all B_20 hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(div_mod_int(A_23),B_20)),B_20) = zero_zero_int) # label(fact_2613_mod__div__trivial) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 939 (all X_2 all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(semiri1621563631at_int,X_2)),hAPP_nat_int(semiri1621563631at_int,Y_2))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2)))) # label(fact_3047_zdvd__int) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 940 (all B_71 all A_82 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_82)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_71)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_82),B_71)))))) # label(fact_2172_add__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 941 (all Ma all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Ma)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)))) # label(fact_1803_nat__0__less__mult__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 942 (all V_22 all V_21 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_21)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_22)) -> hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_21)),hAPP_int_complex(number528085621omplex,V_22)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_21),V_22))))) # label(fact_370_semiring__mult__number__of) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 943 (all P_1 all I_1 (is_int(I_1) -> (hBOOL(hAPP_nat_bool(P_1,hAPP_int_nat(nat,I_1))) <-> (all N_2 (hAPP_nat_int(semiri1621563631at_int,N_2) = I_1 -> hBOOL(hAPP_nat_bool(P_1,N_2)))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),zero_zero_int)) -> hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)))))) # label(fact_3168_split__nat) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 944 (all X_2 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven)) | hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zOdd)))) # label(fact_2903_even__odd__disj) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 945 (all Z_9 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Z_9),hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Z_9),Z_9)) # label(fact_503_mult__2__right) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 946 (all X_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)))))) # label(fact_3918_real__root__gt__zero) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 947 (all B_58 all B_57 all A_66 all C_26 all A_65 (hAPP_int_int(div_mod_int(A_65),C_26) = hAPP_int_int(div_mod_int(A_66),C_26) -> (hAPP_int_int(div_mod_int(B_57),C_26) = hAPP_int_int(div_mod_int(B_58),C_26) -> hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_65),B_57)),C_26) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_66),B_58)),C_26)))) # label(fact_2338_mod__mult__cong) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 948 (all X_1 all Y_1 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,X_1),Y_1)) -> (exists P_4 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_4),X_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_4),Y_1)) & hBOOL(hAPP_nat_bool(prime,P_4)))))) # label(fact_4202_coprime__prime__dvd__ex) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 949 (all V (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),V)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_int_nat(number_number_of_nat,V))))) # label(fact_999_less__0__number__of) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 950 (all C_73 all A_154 all B_130 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_154),B_130)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_73)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_154),C_73)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_130),C_73)))))) # label(fact_1217_mult__right__mono) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 951 (all B_49 all B_48 all A_57 all C_20 all A_56 (hAPP_C498520661umeral(div_mo1740067990umeral(A_56),C_20) = hAPP_C498520661umeral(div_mo1740067990umeral(A_57),C_20) -> (hAPP_C498520661umeral(div_mo1740067990umeral(B_48),C_20) = hAPP_C498520661umeral(div_mo1740067990umeral(B_49),C_20) -> hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_57),B_49)),C_20) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_56),B_48)),C_20)))) # label(fact_2372_mod__add__cong) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 952 (all X_8 all Q_6 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_8),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_8),Q_6)) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_8),hAPP_nat_nat(suc,Q_6))) # label(fact_3013_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 953 (all Ma all N (hAPP_nat_real(semiri132038758t_real,Ma) = hAPP_nat_real(semiri132038758t_real,N) <-> Ma = N)) # label(fact_2834_of__nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 954 (all Lx_3 all Ly_1 all Rx_3 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_3),Rx_3)),Ly_1) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_3),Ly_1)),Rx_3)) # label(fact_161_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 955 (all Z_1 all W (is_int(W) & is_int(Z_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),W)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z_1)) -> Z_1 = W)))) # label(fact_68_zle__antisym) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 956 (all A_60 all B_52 all C_23 hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_60),B_52)),C_23) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_60),C_23)),hAPP_Q1762011733de_int(div_mo231679042de_int(B_52),C_23))),C_23)) # label(fact_2361_mod__add__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 957 (all A_105 A_105 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A_105)) # label(fact_1982_add__0) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 958 (all M hAPP_int_int(abs_abs_int,M) = hAPP_int_int(legacy_zgcd(M),zero_zero_int)) # label(fact_4504_zgcd__0) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 959 (all Sa all Ta all A all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A),B)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Sa),Ta)) -> hBOOL(hAPP_f115998247l_bool(hAPP_P1876494581l_bool(member180897546at_nat,hAPP_P369611207at_nat(hAPP_P402336767at_nat(produc494345619at_nat,hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,A),Sa)),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,B),Ta))),pair_less))))) # label(fact_4539_pair__lessI2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 960 (all X_1 hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),pi))) = hAPP_real_real(tan,X_1)) # label(fact_3609_tan__periodic) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 961 (all X_2 all Y_2 all Z_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_2),Z_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_2),Z_2)))))) # label(fact_1600_real__mult__less__iff1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 962 (all A_219 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_219),A_219)),A_219) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_219),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) # label(fact_480_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 963 (all A_62 all B_54 all C_25 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_62),B_54)),C_25) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_62),hAPP_int_int(div_mod_int(B_54),C_25))),C_25)) # label(fact_2350_mod__add__right__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 964 (all N_1 all M ((M != zero_zero_nat -> hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1)) & (zero_zero_nat = M -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1) = N_1))) # label(fact_3184_add__eq__if) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 965 (all X_1 all Y_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_1),Y_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,X_1),Y_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_1792_diff__square) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 966 (all Z_1 (is_int(Z_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1)) -> hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat,Z_1)) = Z_1) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1)) -> hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat,Z_1)) = zero_zero_int))) # label(fact_3126_int__nat__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 967 (all R_3 all A_3 hAPP_complex_complex(invers1449016382omplex,rcis(R_3,A_3)) = rcis(hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),R_3),hAPP_real_real(uminus_uminus_real,A_3))) # label(fact_4266_rcis__inverse) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 968 (all P all Q all R hAPP_r1134773055l_bool(P,hAPP_real_real(Q,R)) = hAPP_r1134773055l_bool(hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__H_009) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 969 (all B_128 all A_152 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_152),zero_z126310315umeral)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_128)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_152),B_128)),zero_z126310315umeral))))) # label(fact_1224_mult__nonpos__nonneg) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 970 (all N_1 all K_2 all M (hAPP_int_int(legacy_zgcd(K_2),M) = one_one_int -> (one_one_int = hAPP_int_int(legacy_zgcd(N_1),M) -> hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N_1)),M) = one_one_int))) # label(fact_4501_zgcd__zgcd__zmult) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 971 (all Y_2 all X_2 (is_int(X_2) & is_int(Y_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2)) -> (X_2 = Y_2 <-> hAPP_int_nat(nat,Y_2) = hAPP_int_nat(nat,X_2)))))) # label(fact_2725_transfer__nat__int__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 972 (all A_3 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(ln,X_1)),A_3)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(powr,A_3),A_3)),X_1)))))) # label(fact_4475_ln__powr__bound2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 973 (all K_2 all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),M)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),N_1)))))) # label(fact_1667_dvd__diffD1) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 974 (all Y_1 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,one_one_real)),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(ln,Y_1)),Y_1)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(ln,X_1)),X_1)))))) # label(fact_3853_ln__x__over__x__mono) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 975 (all V_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_int_nat(number_number_of_nat,V_2))) -> hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(number_number_of_nat,V_2)),one_one_nat)) = hAPP_int_nat(number_number_of_nat,V_2))) # label(fact_3202_expand__Suc) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 976 (all A_3 all N_1 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N_1)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),N_1))) -> (zero_zero_nat != N_1 -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_3),B_2))))) # label(fact_2286_pow__divides__pow__int) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 977 (all B_90 all A_111 all C_50 (hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_90),A_111) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_50),A_111) -> B_90 = C_50)) # label(fact_1916_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 978 (all A_139 all B_116 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_139),B_116))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_139)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_116))))) # label(fact_1297_zero__less__mult__pos) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 979 (all Y_1 all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_1)))))) # label(fact_1390_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 980 (all K_2 all L_2 all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),L_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_2),L_2)))))) # label(fact_1791_mult__le__mono) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 981 (all N all V (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,N)),hAPP_int_nat(number_number_of_nat,V))) <-> (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))))) & -hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))))) # label(fact_4783_less__Suc__number__of) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 982 (all V_2 all W hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,W)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_2),W))) # label(fact_363_times__numeral__code_I5_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 983 (all N_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1)))) # label(fact_1510_two__realpow__ge__one) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 984 (all N_1 all A_3 all B_2 (one_one_int = hAPP_int_int(legacy_zgcd(A_3),B_2) -> one_one_int = hAPP_int_int(legacy_zgcd(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N_1)),B_2))) # label(fact_4505_zgcd__1__power__left__distrib) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 985 (all X_1 all Y_1 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X_1)),hAPP_real_real(cos,Y_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_1)),hAPP_real_real(sin,Y_1))) = hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),Y_1))) # label(fact_3666_sin__diff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 986 (all C_59 all A_125 all B_104 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_59),A_125)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_59),B_104))) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_59)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_125),B_104))))) # label(fact_1454_mult__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 987 (all A all B (hAPP_real_complex(hAPP_r265291036omplex(complex,A),B) = zero_zero_complex <-> B = zero_zero_real & A = zero_zero_real)) # label(fact_3956_Complex__eq__0) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 988 (all N_1 all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1)) -> hAPP_int_nat(nat,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(nat,Z_1)),N_1))) # label(fact_2761_nat__power__eq) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 989 (all A_163 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,one_one_real),A_163))) # label(fact_1148_one__dvd) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 990 (all Y_1 all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_int_nat(nat,X_1)),hAPP_int_nat(nat,Y_1)) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),Y_1))))) # label(fact_2719_Divides_Otransfer__nat__int__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 991 (all X_2 hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_4866_Ln_Oaux2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 992 (all V_2 hAPP_int_real(number267125858f_real,V_2) = hAPP_int_real(real_int,hAPP_int_int(number_number_of_int,V_2))) # label(fact_4393_real__number__of) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 993 (all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N)) -> hAPP_f957591787ol_nat(finite_card_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),N)))) = hAPP_int_nat(nat,N))) # label(fact_4738_card__bdd__int__set__l) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 994 (all B all P_5 all A (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(wset(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),one_one_int)),P_5))) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(wset(A),P_5)))))) # label(fact_2237_wset__subset) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 995 (all X_2 all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,X_2)),hAPP_int_real(number267125858f_real,Y_2))))) # label(fact_94_le__number__of) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 996 (all Y_1 all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1)) -> hAPP_int_nat(nat,hAPP_int_int(nat_tsub(X_1),Y_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat,X_1)),hAPP_int_nat(nat,Y_1))))) # label(fact_2784_Nat__Transfer_Otransfer__nat__int__functions_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 997 (all Z_2 (hAPP_complex_complex(cnj,Z_2) = zero_zero_complex <-> Z_2 = zero_zero_complex)) # label(fact_4064_complex__cnj__zero__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 998 (all A all B (hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A),B) = zero_zero_complex <-> B = zero_zero_complex | zero_zero_complex = A)) # label(fact_1034_mult__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 999 (all X_1 all Y_1 hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,Y_1)),hAPP_real_real(cos,X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,Y_1)),hAPP_real_real(sin,X_1)))) # label(fact_3681_cos__diff2) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 1000 (all A all B (hAPP_int_int(div_mod_int(B),A) = zero_zero_int <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),B)))) # label(fact_2433_dvd__eq__mod__eq__0) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 1001 (all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arcsin,Y_1)),pi)) & Y_1 = hAPP_real_real(sin,hAPP_real_real(arcsin,Y_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arcsin,Y_1)))))) # label(fact_3744_arcsin__pi) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 1002 (all A_32 all B_28 (B_28 != zero_z891286103de_int -> A_32 = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_32),B_28)),B_28))) # label(fact_2572_div__mult__self2__is__id) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 1003 (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2)) -> hBOOL(hAPP_real_bool(deriv_real(sqrt,X_2),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,X_2))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_3876_DERIV__real__sqrt) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 1004 (all A_8 all N_14 all B_7 (hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_8),hAPP_nat_nat(suc,N_14)) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_7),hAPP_nat_nat(suc,N_14)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_8)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_7)) -> B_7 = A_8)))) # label(fact_3114_power__inject__base) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 1005 (all N all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),Ma)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(real_nat,Ma))))) # label(fact_3647_real__of__nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 1006 (all A_127 all C_61 all B_106 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_127),C_61)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_106),C_61))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_61)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_127),B_106))))) # label(fact_1439_mult__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 1007 (all Y_2 all X_2 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zOdd)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),Y_2)),zOdd))))) # label(fact_3098_even__minus__odd) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 1008 (all X_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_real_real(abs_abs_real,X_1))))) # label(fact_3397_abs__add__one__gt__zero) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 1009 (all X_1 all D_3 (is_int(D_3) -> (zero_zero_int != D_3 -> hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,X_1)),hAPP_int_real(real_int,D_3)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),D_3))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,hAPP_int_int(div_mod_int(X_1),D_3))),hAPP_int_real(real_int,D_3)))))) # label(fact_4465_real__of__int__div__aux) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 1010 (all N_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),N_1))) # label(fact_1728_le__refl) # label(axiom) # label(non_clause). [assumption]. 5.34/5.43 1011 (all Y_1 all X_1 exists R_1 exists A_2 (X_1 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(cos,A_2)) & hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(sin,A_2)) = Y_1)) # label(fact_4543_polar__Ex) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1012 (all X_2 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_2),N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),X_2)) | zero_zero_nat = N)) # label(fact_921_zero__less__power__nat__eq) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1013 (all P all Q all R hAPP_r2019696015l_real(hAPP_f320596107l_real(hAPP_f850851693l_real(cOMBS_193881978l_real,P),Q),R) = hAPP_f617735571l_real(hAPP_r712953929l_real(P,R),hAPP_r1250527377l_real(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__R) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1014 (all S (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S)) -> (exists K hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,S),hAPP_n1699378549t_bool(ord_lessThan_nat,K)))))) # label(fact_4954_finite__nat__bounded) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1015 (all B_2 all A_3 all A_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),A_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_5),B_2)))))) # label(fact_2623_zdiv__mono1) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1016 (all X_1 hAPP_real_real(exp_real,X_1) = hAPP_real_real(abs_abs_real,hAPP_real_real(exp_real,X_1))) # label(fact_3827_abs__exp__cancel) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1017 (all P all X_1 (hBOOL(hAPP_f54304608l_bool(fEx_nat,P)) | -hBOOL(hAPP_nat_bool(P,X_1)))) # label(help_fEx_1_1_fEx_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1018 (all M all N_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)),hAPP_nat_nat(suc,M)))) # label(fact_3068_diff__less__Suc) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1019 (all Ta all D all D_1 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,D),D_1)) -> (all X all K (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,D),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X),Ta))) <-> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,D),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,K),D_1))),Ta))))))) # label(fact_1814_inf__period_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1020 (all X ((-hBOOL(hAPP_nat_bool(even_odd_even_nat,X)) -> zero_zero_real = hAPP_nat_real(cos_coeff,X)) & (hBOOL(hAPP_nat_bool(even_odd_even_nat,X)) -> hAPP_nat_real(cos_coeff,X) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,X)))))) # label(fact_3797_cos__coeff__def) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1021 (all A_223 A_223 = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,pls)),A_223)) # label(fact_416_add__numeral__0) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1022 (all N_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,N_1))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,N_1)))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,N_1)))))) # label(fact_1633_number__of1) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1023 (all X_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> hAPP_real_real(hAPP_n546711566l_real(root,N_1),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1)) = X_1))) # label(fact_3934_real__root__pos2) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1024 (all N hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),N))) = N) # label(fact_4763_card__Collect__less__nat) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1025 (all X_2 all Y_2 (zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Y_2) <-> hAPP_real_real(uminus_uminus_real,X_2) = Y_2)) # label(fact_3483_real__add__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1026 (all A_225 all M_28 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_225),one_one_complex)),M_28) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_225),M_28)),M_28)) # label(fact_396_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1027 (all P all Q all R hAPP_f351634443t_bool(hAPP_r1392521531t_bool(P,R),Q) = hAPP_r632163725t_bool(hAPP_f956464319t_bool(hAPP_f422375781t_bool(cOMBC_243356514t_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Int__Oint_Mtc__fun_It) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1028 (all X_1 hAPP_int_int(abs_abs_int,X_1) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),zero_zero_int)) # label(fact_5004_gcd__0__int) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1029 (all Y_1 all X_1 all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Y_1),zero_zero_int),P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(multInv(P_3),X_1)),hAPP_int_int(multInv(P_3),Y_1)),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),Y_1),P_3)))))))) # label(fact_2144_MultInv__prop5) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1030 (all F all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)) -> ((all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),B)) -> hBOOL(hAPP_real_bool(isCont_real_real(F),X)))) -> ((all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),X)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),B)) -> hBOOL(hAPP_real_bool(deriv_real(F,X),zero_zero_real)))) -> (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),B)) -> hAPP_real_real(F,A) = hAPP_real_real(F,X))))))) # label(fact_5178_DERIV__isconst1) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1031 (all B_1_1 all B_2_1 is_bool(hAPP_f448129468l_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1032 (all W_17 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,W_17)),hAPP_int_real(number267125858f_real,W_17)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,W_17)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_18_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1033 (all N_1 all A_3 all B_2 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),B_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,P_3),N_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,P_3),N_1)),A_3)))))) # label(fact_727_zprime__power__zdvd__cancel__right) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1034 (all B_1_1 all B_2_1 is_bool(hAPP_real_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__RealDef__Oreal_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1035 (all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),hAPP_int_int(bit1,K_1))))) # label(fact_271_rel__simps_I22_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1036 (all A_76 all B_65 hAPP_Q1762011733de_int(div_mo231679042de_int(A_76),B_65) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(div_mo231679042de_int(A_76),B_65)),B_65)) # label(fact_2301_mod__mod__trivial) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1037 (all N_19 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,zero_zero_complex),hAPP_nat_nat(suc,N_19)) = zero_zero_complex) # label(fact_2995_power__0__Suc) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1038 (all N_1 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N_1)),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,N_1)),one_one_real))) -> natfloor(X_1) = N_1))) # label(fact_3704_natfloor__eq) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1039 (all N_1 N_1 = hAPP_C498520661umeral(hAPP_C1594335432umeral(minus_1690775515umeral,code_S1047413653umeral(N_1)),one_on1645066479umeral)) # label(fact_4084_Suc__code__numeral__minus__one) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1040 (all C_57 all A_123 all B_102 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_57),A_123)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_57),B_102))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_57)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_123),B_102))))) # label(fact_1462_mult__left__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1041 (all N_1 (-hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1)) -> (exists X hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X)),one_one_nat)))) # label(fact_3794_odd__square) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1042 (all X_2 all Y_2 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_2),Y_2)),zOdd)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zOdd)))) # label(fact_2554_odd__mult__odd__prop) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1043 (all W_16 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_int_real(number267125858f_real,W_16)) = hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit1,pls)),W_16))) # label(fact_51_add__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1044 (all B_1_1 all B_2_1 is_bool(hAPP_f1501066425l_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__fun_Itc__RealDef__Oreal_Mtc__HOL__Obool_J_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1045 (all Z_5 (hBOOL(hAPP_int_bool(nat_neg,Z_5)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_5),zero_zero_int)))) # label(fact_3273_neg__def) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1046 (all C_38 all A_95 all B_78 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_95),B_78)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_38),A_95)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_38),B_78))))) # label(fact_2055_add__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1047 (all P all Q all R hAPP_i1524277240nt_int(P,hAPP_int_int(Q,R)) = hAPP_i1524277240nt_int(hAPP_f883232462nt_int(hAPP_f2002277285nt_int(cOMBB_1846624359nt_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1048 (all P_1 all P_2 all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D)) -> ((all X all K (is_int(K) & is_int(X) -> (hBOOL(hAPP_int_bool(P_2,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D)))) <-> hBOOL(hAPP_int_bool(P_2,X))))) -> ((exists Z all X (is_int(X) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z),X)) -> (hBOOL(hAPP_int_bool(P_1,X)) <-> hBOOL(hAPP_int_bool(P_2,X)))))) -> ((exists X1 hBOOL(hAPP_int_bool(P_2,X1))) -> (exists X1 (hBOOL(hAPP_int_bool(P_1,X1)) & is_int(X1)))))))) # label(fact_4005_plusinfinity) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1049 (all Z_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_int_nat(nat,Z_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_2)))) # label(fact_2749_zero__less__nat__eq) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1050 (all A_91 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_91),one_on1645066479umeral) = A_91) # label(fact_2103_mult_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1051 (all X_44 all Y_36 all Z_15 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_44),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Y_36),Z_15)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_44),Y_36)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_44),Z_15))) # label(fact_324_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1052 (all A_135 -hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_135),A_135)),zero_zero_rat))) # label(fact_1331_not__square__less__zero) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1053 (all Y_3 all X_3 ((exists A_2 ((all N_2 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,hAPP_nat_rat(X_3,N_2))),A_2))) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_2)))) -> (hBOOL(vanishes(Y_3)) -> hBOOL(vanishes(hAPP_f2062659861at_rat(hAPP_f1131305186at_rat(cOMBS_nat_rat_rat,hAPP_f1830834240at_rat(hAPP_f1073644437at_rat(cOMBB_1926947at_nat,times_times_rat),X_3)),Y_3)))))) # label(fact_5126_vanishes__mult__bounded) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1054 (all A (hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially)) -> ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(A,N_2)))) -> ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A,hAPP_nat_nat(suc,N_2))),hAPP_nat_real(A,N_2)))) -> hBOOL(tendsto_nat_real(hAPP_f292751352t_real(hAPP_f1607377819t_real(cOMBB_1533963633al_nat,hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A))),hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,ord_at4362885an_nat(zero_zero_nat)),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),sequentially)))))) # label(fact_5060_summable__Leibniz_H_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1055 (all L all U hAPP_f22106695ol_nat(finite_card_nat,hAPP_n1699378549t_bool(ord_at238088361st_nat(L),U)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,U)),L)) # label(fact_5078_card__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1056 (all X_2 all Y_2 all B (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),B)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),Y_2)) <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B),X_2)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B),Y_2)))))) # label(fact_882_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1057 (all A all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),P_5)) -> hAPP_int_nat(nat,A) = hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(bnorRset(A),P_5))))) # label(fact_4289_Bnor__prime) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1058 (all X_29 all Y_23 (is_int(Y_23) & is_int(X_29) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_29),Y_23)) -> (X_29 != Y_23 -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_29),Y_23)))))) # label(fact_1011_order__le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1059 (all Ma all K_1 all N (hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,K_1),N) = one_one_nat -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),Ma))))) # label(fact_4367_coprime__dvd__mult__iff__nat) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1060 (all B_58 all B_57 all A_66 all C_26 all A_65 (hAPP_nat_nat(div_mod_nat(A_66),C_26) = hAPP_nat_nat(div_mod_nat(A_65),C_26) -> (hAPP_nat_nat(div_mod_nat(B_58),C_26) = hAPP_nat_nat(div_mod_nat(B_57),C_26) -> hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_65),B_57)),C_26) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_66),B_58)),C_26)))) # label(fact_2339_mod__mult__cong) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1061 (all V_11 all W_9 hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_11),W_9)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_11)),hAPP_int_int(number_number_of_int,W_9))) # label(fact_441_number__of__add) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1062 (all X_1 (zero_zero_real != X_1 -> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(inverse_inverse_real,X_1)),X_1) = one_one_real)) # label(fact_3838_real__mult__inverse__left) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1063 (all V_13 all W_11 all Z_13 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_13)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,W_11)),Z_13)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_13),W_11))),Z_13)) # label(fact_434_add__number__of__left) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1064 (all P all Q all R hAPP_f1356864277nt_int(hAPP_i744251628nt_int(P,R),Q) = hAPP_int_fun_int_int(hAPP_f1760145644nt_int(hAPP_f808647893nt_int(cOMBC_736024425nt_int,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1065 (all X_2 archim791455193or_rat(X_2) = the_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_r1900510169t_bool(hAPP_f1480506641t_bool(cOMBC_int_rat_bool,hAPP_f601898971t_bool(hAPP_f857296063t_bool(cOMBB_214503962ol_int,ord_less_eq_rat),ring_1_of_int_rat)),X_2))),hAPP_f1241350704t_bool(hAPP_f1131424617t_bool(cOMBB_rat_bool_int,hAPP_r2115560837t_bool(ord_less_rat,X_2)),hAPP_f1900950721nt_rat(hAPP_f1270196437nt_rat(cOMBB_int_rat_int,ring_1_of_int_rat),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,plus_plus_int),one_one_int)))))) # label(fact_4890_floor__rat__def) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1066 (all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(semiri984289939at_nat,N))))) # label(fact_3149_of__nat__0__less__iff) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1067 (all X_1 all Y_1 hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_1)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X_1),Y_1))) # label(fact_3078_Divides_Otransfer__int__nat__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1068 (all N_56 all A_206 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_206)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_206),N_56))))) # label(fact_649_zero__le__power) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1069 (all P all Q all R hAPP_int_real(P,hAPP_int_int(Q,R)) = hAPP_int_real(hAPP_f563293398t_real(hAPP_f134192053t_real(cOMBB_int_real_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__RealDef__Oreal_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1070 (all P_1 ((all X hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,X)))) <-> (all X (is_int(X) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> hBOOL(hAPP_int_bool(P_1,X))))))) # label(fact_2987_transfer__int__nat__quantifiers_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1071 (all Y_1 all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1)) -> hAPP_nat_nat(div_mod_nat(hAPP_int_nat(nat,X_1)),hAPP_int_nat(nat,Y_1)) = hAPP_int_nat(nat,hAPP_int_int(div_mod_int(X_1),Y_1))))) # label(fact_2762_Divides_Otransfer__nat__int__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1072 (all X_8 all Q_6 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_8),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_8),Q_6)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_8),hAPP_nat_nat(suc,Q_6))) # label(fact_3018_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1073 (all A_105 (is_int(A_105) -> A_105 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),A_105))) # label(fact_1981_add__0) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1074 (all Lx_5 all Ly_3 all Rx_5 all Ry_3 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_5),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_5),Ly_3)),Ry_3)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_5),Ly_3)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_5),Ry_3))) # label(fact_145_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1075 (all A_231 all N_68 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_231),N_68)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_231),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_68))) # label(fact_277_power__even__eq) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1076 (all X_2 all Y_2 (ii = hAPP_real_complex(hAPP_r265291036omplex(complex,X_2),Y_2) <-> X_2 = zero_zero_real & one_one_real = Y_2)) # label(fact_3979_Complex__eq__i) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1077 (all X_1 hAPP_int_int(abs_abs_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(abs_abs_int,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) # label(fact_3447_abs__power3__distrib) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1078 (all X_1 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),X_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)))),one_one_int),P_3))))) # label(fact_2768_Little__Fermat) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1079 (all X_2 (zero_zero_real = X_2 <-> X_2 = zero_zero_real)) # label(fact_1898_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1080 (all N_57 all A_211 all B_157 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_211),B_157)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_211)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_57)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_211),N_57)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_157),N_57))))))) # label(fact_606_power__strict__mono) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1081 (all A all B all C (C = B <-> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),C) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),B))) # label(fact_1941_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1082 (all X_2 all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(number_number_of_int,X_2)),hAPP_int_int(number_number_of_int,Y_2))) <-> hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,Y_2)),hAPP_int_int(number_number_of_int,X_2)) = zero_zero_int)) # label(fact_2457_zdvd__iff__zmod__eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1083 (all X_2 all Y_2 all B (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),B)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),Y_2)) <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B),X_2)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B),Y_2)))))) # label(fact_538_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1084 (all C_39 all D_7 all A_96 all B_79 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_96),B_79)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_39),D_7)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_96),C_39)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_79),D_7)))))) # label(fact_2049_add__strict__mono) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1085 (all I all J_2 -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J_2)),I))) # label(fact_1766_not__add__less1) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1086 (all P all Q all R hAPP_i1584592887nt_int(P,hAPP_int_int(Q,R)) = hAPP_i1584592887nt_int(hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__prod_Itc__I) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1087 (all L (is_int(L) -> (L = pls <-> pls = hAPP_int_int(bit0,L)))) # label(fact_353_rel__simps_I38_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1088 (all C all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),B)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),C)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B),C)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B),A)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,C),A))))))) # label(fact_2551_div__dvd__div) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1089 (all B_137 all A_161 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_137)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_161)) | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_161),zero_zero_rat)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_137),zero_zero_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_161),B_137))))) # label(fact_1181_split__mult__pos__le) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1090 (all N_56 all A_206 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_206)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_206),N_56))))) # label(fact_648_zero__le__power) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1091 (all M_9 all N_23 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(semiri1619134803umeral,M_9)),hAPP_n1108039445umeral(semiri1619134803umeral,N_23))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_23)))) # label(fact_2943_of__nat__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1092 (all A_44 all B_37 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_44),B_37)) -> zero_z891286103de_int = hAPP_Q1762011733de_int(div_mo231679042de_int(B_37),A_44))) # label(fact_2428_dvd__imp__mod__0) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1093 (all X_1 hAPP_real_real(abs_abs_real,X_1) = hAPP_real_real(sqrt,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3401_real__sqrt__abs) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1094 (all A all B all K_1 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ma)) -> (hAPP_int_int(legacy_zgcd(K_1),Ma) = one_one_int -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),K_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),K_1)),Ma)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma)))))) # label(fact_4524_zcong__cancel) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1095 (all C_81 all A_167 all B_142 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_167),B_142)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_167),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_142),C_81))))) # label(fact_1120_dvd__mult2) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1096 (all K_1 (hBOOL(hAPP_rat_bool(iszero_rat,hAPP_int_rat(number_number_of_rat,K_1))) <-> hBOOL(hAPP_int_bool(iszero_int,hAPP_int_int(number_number_of_int,K_1))))) # label(fact_4635_iszero__rat) # label(axiom) # label(non_clause). [assumption]. 5.39/5.43 1097 (all Ta all D all D_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),D_1)) -> (all X all K (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Ta))) <-> -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,K),D_1))),Ta))))))) # label(fact_1817_inf__period_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1098 (all D_3 all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),A_3)))) # label(fact_4010_coprime__rmul2) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1099 (all Y_9 all Z_6 all X_13 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_6),X_13)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_6),Y_9)) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_13),Y_9)),Z_6) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_13),Z_6)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,Y_9),Z_6))))) # label(fact_2611_div__add) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1100 (all B_1_1 all B_2_1 all B_3_1 is_bool(tendsto_real_real(B_1_1,B_2_1,B_3_1))) # label(gsy_c_Limits_Otendsto_000tc__RealDef__Oreal_000tc__RealDef__Oreal) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1101 (all X_1 all Y_1 all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),Y_1),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(multInv(P_3),X_1)),hAPP_int_int(multInv(P_3),Y_1)),P_3))))) # label(fact_2141_MultInv__prop1) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1102 (all A_3 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(div_mod_int(A_3),B_2)),zero_zero_int)))) # label(fact_2450_neg__mod__sign) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1103 (all B_94 all C_52 all A_115 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_115)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_94),C_52)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_94),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_115),C_52)))))) # label(fact_1875_add__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1104 (all A_177 zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_177),zero_zero_nat)) # label(fact_1037_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1105 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),zero_zero_int)) -> hAPP_P1175774780nt_int(product_fst_int_int,hAPP_P1975530577nt_int(negateSnd,posDivAlg(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2)))) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)))) # label(fact_4228_div__neg__neg) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1106 (all M_8 all N_22 hAPP_n522471361de_int(semiri1424489471de_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_8),N_22)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_n522471361de_int(semiri1424489471de_int,M_8)),hAPP_n522471361de_int(semiri1424489471de_int,N_22))) # label(fact_2957_of__nat__add) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1107 (all M hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),zero_zero_nat) = M) # label(fact_1639_minus__nat_Odiff__0) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1108 (all N_1 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,M),N_1)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N_1),M))))) # label(fact_580_zdvd__not__zless) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1109 (all A_210 zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,zero_zero_nat),A_210)) # label(fact_609_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1110 (all P all Q all R hAPP_f41603166t_bool(P,hAPP_r373117745t_bool(Q,R)) = hAPP_r481142414t_bool(hAPP_f1341793266t_bool(hAPP_f571820015t_bool(cOMBB_1159313332l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Nat__Onat_Mtc__HOL_) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1111 (all C_42 all A_99 all B_82 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_99),B_82)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_99),C_42)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_82),C_42))))) # label(fact_2021_add__right__mono) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1112 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),X_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X_1))),X_1)))) # label(fact_3456_ln__add__one__self__le__self2) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1113 (all X_1 exists N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_nat_real(real_nat,N_2)))) # label(fact_3951_reals__Archimedean2) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1114 (all A (A = zero_zero_real <-> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),A) = zero_zero_real)) # label(fact_1978_double__zero__sym) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1115 (all N_1 all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M)) -> M = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)))) # label(fact_1676_le__add__diff__inverse) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1116 (all B_97 all A_118 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_118),zero_zero_rat)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_97),zero_zero_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_118),B_97)),zero_zero_rat))))) # label(fact_1861_add__nonpos__neg) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1117 (all X_1 hAPP_real_real(uminus_uminus_real,hAPP_real_real(arctan,X_1)) = hAPP_real_real(arctan,hAPP_real_real(uminus_uminus_real,X_1))) # label(fact_3480_arctan__minus) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1118 (all Z_11 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Z_11),Z_11) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_11)) # label(fact_494_mult__2) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1119 (all L all U hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(ord_at875362053st_int(L),U)))) # label(fact_5066_finite__atLeastAtMost__int) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1120 (all B_71 all A_82 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_82)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_71)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_82),B_71)))))) # label(fact_2174_add__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1121 (all B_96 all A_117 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_117),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_96),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_117),B_96)),zero_zero_int))))) # label(fact_1863_add__neg__nonpos) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1122 (all Ma all N hAPP_nat_nat(nat_case_nat(hAPP_nat_nat(suc,N),hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,suc),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,ord_max_nat),N))),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,Ma),hAPP_nat_nat(suc,N))) # label(fact_5193_max__Suc2) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1123 (all P all Q all R hAPP_real_real(P,hAPP_nat_real(Q,R)) = hAPP_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__Nat__Ona) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1124 (all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,pls)),Y_2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),hAPP_int_int(number_number_of_int,Y_2))))) # label(fact_293_le__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1125 (all N_67 all A_218 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_218)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_218),N_67)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_218),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_218),N_67)))))) # label(fact_520_power__less__power__Suc) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1126 (all A_3 all P_3 (is_int(A_3) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),A_3)),one_one_int),P_3)) -> A_3 = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int) | one_one_int = A_3)))))) # label(fact_1610_zcong__square__zless) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1127 (all X_38 all Y_30 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_38),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_30),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),Y_30)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,X_38),Y_30))))) # label(fact_786_power2__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1128 (all Ma all N (zero_zero_nat != N & Ma = zero_zero_nat <-> zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Ma),N))) # label(fact_1593_nat__power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1129 (all A_237 all B_166 all C_97 all D_20 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_237),C_97)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_166),D_20)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_237),B_166)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_97),D_20))) # label(fact_195_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1130 (all P_1 all Ma all N ((hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N),Ma))) -> ((N = Ma -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N),Ma))) -> ((hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),Ma)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N),Ma))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N),Ma)))))) # label(fact_1689_nat__less__cases) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1131 (all A_237 all B_166 all C_97 all D_20 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_237),C_97)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_166),D_20)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_237),B_166)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_97),D_20))) # label(fact_197_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1132 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),M)),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N_1))) # label(fact_4357_gcd__diff2__nat) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1133 (all K_2 all L_2 all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_2),L_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),L_2)))))) # label(fact_1760_add__less__mono) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1134 (all U_1 all X_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,U_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_3496_realpow__square__minus__le) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1135 (all C_83 all D_16 all A_169 all B_144 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_169),B_144)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,C_83),D_16)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_169),C_83)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_144),D_16)))))) # label(fact_1106_mult__dvd__mono) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1136 (all R_3 of_real_complex(R_3) = hAPP_real_complex(hAPP_r265291036omplex(complex,R_3),zero_zero_real)) # label(fact_3987_complex__of__real__def) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1137 (all M_26 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,one_one_complex),one_one_complex)),M_26) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,M_26),M_26)) # label(fact_410_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1138 (all Z_1 all W hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z_1) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),W)) # label(fact_261_zmult__commute) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1139 (all S ((all M_1 exists N_2 (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,N_2),S)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2)))) <-> -hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S)))) # label(fact_4607_infinite__nat__iff__unbounded__le) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1140 (all W_6 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,zero_zero_real),hAPP_int_real(number267125858f_real,W_6))),hAPP_int_real(number267125858f_real,W_6)) = hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,W_6))) # label(fact_743_number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1141 (all A all B (zero_zero_real = B | A = zero_zero_real <-> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),B) = zero_zero_real)) # label(fact_1033_mult__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1142 (all B_127 all A_151 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_151)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_127),zero_z126310315umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_127),A_151)),zero_z126310315umeral))))) # label(fact_1230_mult__nonneg__nonpos2) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1143 (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sqrt,X_2)),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),zero_zero_real)))) # label(fact_3324_real__sqrt__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1144 (all X_1 (zero_zero_real != X_1 -> hAPP_real_real(arctan,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),X_1)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sgn_sgn_real,X_1)),pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_real_real(arctan,X_1)))) # label(fact_3557_arctan__inverse) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1145 (all V_8 all W_4 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(number_number_of_int,V_8)),hAPP_int_int(number_number_of_int,W_4)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,V_8),W_4))) # label(fact_1516_number__of__diff) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1146 (all I all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),N_1)) -> I = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),I)))) # label(fact_1652_diff__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1147 (all A_166 all B_141 hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_166),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_141),A_166)))) # label(fact_1127_dvd__triv__right) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1148 (all F all X_2 all F_1 all N all Ma ((all R_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,R_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),N))) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),R_1)) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(F,R_1),X_2),hAPP_real_real(hAPP_n546711566l_real(F_1,R_1),X_2))))) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_f2119584768l_real(hAPP_f59594445l_real(cOMBC_2010481033l_real,hAPP_f1523861863l_real(hAPP_f1067681227l_real(cOMBB_1934313333l_real,big_co604158596t_real),hAPP_f42270917t_real(cOMBC_nat_real_real,F))),hAPP_n1699378549t_bool(ord_at4362885an_nat(Ma),N)),X_2),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,F_1),X_2)),hAPP_n1699378549t_bool(ord_at4362885an_nat(Ma),N)))))) # label(fact_4942_DERIV__sumr) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1149 (all P all Q all R hAPP_real_bool(hAPP_i688218208l_bool(P,R),Q) = hAPP_int_bool(hAPP_r481142414t_bool(hAPP_f1845081853t_bool(cOMBC_int_real_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__RealDef__Oreal_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1150 (all A (hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially)) -> (hBOOL(monoseq_real(A)) -> hBOOL(tendsto_nat_real(hAPP_f292751352t_real(hAPP_f1607377819t_real(cOMBB_1533963633al_nat,hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A))),hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,ord_at4362885an_nat(zero_zero_nat)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),sequentially))))) # label(fact_5031_summable__Leibniz_I5_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1151 (all A_179 all B_151 (zero_zero_rat = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_179),B_151) -> zero_zero_rat = B_151 | A_179 = zero_zero_rat)) # label(fact_1024_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1152 (all A all X_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),archim1246769320r_real(X_2))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,A)),one_one_real)),X_2)))) # label(fact_4467_less__floor__eq) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1153 (all N all Diff all F all H (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H)) -> (F = hAPP_n546711566l_real(Diff,zero_zero_nat) -> ((all M_1 all T (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),T)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),H)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N)) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),T),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T))))) -> (exists T (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,H))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),N))) = hAPP_real_real(F,H) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),H)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),T)))))))) # label(fact_4936_Maclaurin2) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1154 (all N all X_2 all Diff all F (hAPP_n546711566l_real(Diff,zero_zero_nat) = F -> ((all M_1 all X hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),X),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),X)))) -> (exists T (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_2))) & hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,X_2))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),N))) = hAPP_real_real(F,X_2)))))) # label(fact_4939_Maclaurin__all__le) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1155 (all M all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),M)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),M))))) # label(fact_1393_zcong__not__zero) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1156 (all A all N (zero_zero_nat != N & zero_zero_nat = A <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A),N) = zero_zero_nat)) # label(fact_547_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1157 (all N all K_1 all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),Ma)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N)) -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),K_1) <-> N = Ma)))) # label(fact_1651_eq__diff__iff) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1158 (all X_2 all Y_2 (is_int(X_2) & is_int(Y_2) -> (Y_2 != zero_zero_int | zero_zero_int != X_2 <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_2),X_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_2),Y_2))))))) # label(fact_738_sum__squares__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1159 (all A all B all C (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C),zero_zero_rat)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B),A)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),A)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),B)))))) # label(fact_1289_mult__less__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1160 (all N hAPP_real_real(uminus_uminus_real,hAPP_nat_real(sin_coeff,N)) = hAPP_nat_real(diffs_real(cos_coeff),N)) # label(fact_4877_cos__fdiffs2) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1161 (all N_27 all Y_8 all X_12 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_8),X_12)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X_12),Y_8)),N_27) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_12),N_27)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_8),N_27)))) # label(fact_2620_div__power) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1162 (all A_215 all M_24 all N_64 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_24),N_64)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_215),M_24)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_215),N_64))))) # label(fact_553_le__imp__power__dvd) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1163 (all A_219 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_219),A_219)),A_219) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_219),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) # label(fact_484_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1164 (all N all X_2 (zero_zero_real != X_2 -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (exists T (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(abs_abs_real,T))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_2))) & hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(power_power_real,X_2))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(exp_real,T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),N))) = hAPP_real_real(exp_real,X_2)))))) # label(fact_4924_Maclaurin__exp__lt) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1165 (all K_2 all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N_1),K_2))))) # label(fact_2716_div__le__mono) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1166 (all A_231 all N_68 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_231),N_68)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_231),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_68))) # label(fact_280_power__even__eq) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1167 (all X_1 all Y_1 hAPP_complex_real(im,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_complex_real(im,X_1)),hAPP_complex_real(im,Y_1))) # label(fact_4097_Im_Oadd) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1168 (all Y_2 all X_2 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zOdd)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zOdd)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_2),Y_2)),zOdd))))) # label(fact_2553_EvenOdd_Oodd__times__odd) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1169 (all X_2 (hBOOL(hAPP_int_bool(even_odd_even_int,X_2)) <-> hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(uminus_uminus_int,X_2))))) # label(fact_3526_even__neg) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1170 (all P_1 all Y_2 all X_2 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_2),Y_2))))) -> ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_2),zero_zero_int)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_2),hAPP_int_int(uminus_uminus_int,Y_2)))))) -> ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(uminus_uminus_int,X_2)),Y_2))))) -> ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_2),zero_zero_int)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(uminus_uminus_int,X_2)),hAPP_int_int(uminus_uminus_int,Y_2)))))) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_2),Y_2)))))))) # label(fact_5012_gcd__cases__int) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1171 (all A all B (hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(posDivAlg_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A),B))) -> (-(hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),zero_zero_int))) -> hAPP_P1975530577nt_int(adjust(B),posDivAlg(A,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B))) = posDivAlg(A,B)) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),zero_zero_int)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),A) = posDivAlg(A,B)))) # label(fact_2531_posDivAlg_Opsimps) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1172 (exists K_4 ((all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(re,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X)),K_4)))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),K_4)))) # label(fact_4140_Re_Ononneg__bounded) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1173 (all X_1 X_1 = hAPP_real_real(hAPP_n546711566l_real(root,hAPP_nat_nat(suc,zero_zero_nat)),X_1)) # label(fact_3895_real__root__Suc__0) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1174 (all B_70 all A_81 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_81),zero_z891286103de_int)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_70),zero_z891286103de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_81),B_70)),zero_z891286103de_int))))) # label(fact_2188_add__neg__neg) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1175 (all Ma hAPP_i1948725293t_bool(norRRset,Ma) = hAPP_i1948725293t_bool(bnorRset(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Ma),one_one_int)),Ma)) # label(fact_4574_norRRset__def) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1176 (all X_2 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) <-> X_2 != zero_zero_real | Y_2 != zero_zero_real)) # label(fact_855_sum__power2__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1177 (all Ma all N (N = Ma <-> hAPP_nat_nat(semiri984289939at_nat,Ma) = hAPP_nat_nat(semiri984289939at_nat,N))) # label(fact_2832_of__nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1178 (all A_195 all N_47 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_195),N_47)),A_195) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_195),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_195),N_47))) # label(fact_805_power__commutes) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1179 (all X_1 all Y_1 all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_1),Z_1)),Z_1))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),Z_1)),Y_1))))) # label(fact_2694_div__prop2) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1180 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(cos,X_1)))))) # label(fact_3587_cos__gt__zero) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1181 (all N_41 all A_190 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,one_one_rat),A_190)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,one_one_rat),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_190),N_41))))) # label(fact_865_one__le__power) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1182 (all X_1 all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2))) -> -(hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),A_3)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),B_2))))) # label(fact_4314_gcd__semilattice__nat_Ole__infE) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1183 (all M all N_1 all K_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),K_2))) # label(fact_1730_nat__mult__assoc) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1184 (all X_1 all A_3 all B_2 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2) = one_one_nat -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),A_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),B_2)) -> X_1 = one_one_nat)))) # label(fact_4348_coprime__common__divisor__nat) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1185 (all W_17 hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,hAPP_i769753017umeral(number1443263063umeral,W_17)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_i769753017umeral(number1443263063umeral,W_17)),hAPP_i769753017umeral(number1443263063umeral,W_17))) # label(fact_14_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1186 (all M all A_3 (is_int(A_3) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),M)) -> zero_zero_int = A_3))))) # label(fact_1487_zcong__zless__0) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1187 (all N_1 (one_one_nat != N_1 -> (exists P_4 (hBOOL(hAPP_nat_bool(prime,P_4)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_4),N_1)))))) # label(fact_4204_prime__factor) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1188 (all B_2 all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_3)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,archim1246769320r_real(A_3)),archim1246769320r_real(B_2))),archim1246769320r_real(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_3),B_2))))))) # label(fact_4033_le__mult__floor) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1189 (all X_2 (is_int(X_2) -> (one_one_int = X_2 <-> one_one_int = X_2))) # label(fact_1952_one__reorient) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1190 (all X_1 hAPP_real_real(ln,hAPP_real_real(exp_real,X_1)) = X_1) # label(fact_3821_ln__exp) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1191 (all N_1 all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),M)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),hAPP_nat_nat(fact_fact_nat,N_1)))))) # label(fact_3811_dvd__fact__nat) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1192 (all P_1 all A0 all A1 (hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(negDivAlg_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A0),A1))) -> ((all A_2 all B_4 (is_int(A_2) & is_int(B_4) -> (hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(negDivAlg_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_2),B_4))) -> ((-(hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_4),zero_zero_int)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_2),B_4)))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_4)))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A_2),B_4)))))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A0),A1))))) # label(fact_2705_negDivAlg_Opinduct) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1193 (all P all Q all R hAPP_f291087797t_bool(P,hAPP_P1668131738nt_int(Q,R)) = hAPP_P2053772734t_bool(hAPP_f1198736473t_bool(hAPP_f2042071795t_bool(cOMBB_773384995nt_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__I) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1194 (all X_49 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_49),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_49),X_49)) # label(fact_30_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1195 (all K_2 all M all N_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,M),N_1))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),N_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M))))) # label(fact_1538_zdvd__zdiffD) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1196 (all P all Q all R hAPP_nat_real(hAPP_f73570213t_real(hAPP_f918606171t_real(cOMBB_64450052al_nat,P),Q),R) = hAPP_P731461727l_real(P,hAPP_n332733730l_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1197 (all C_43 all A_100 all B_83 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_100),B_83)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_43),A_100)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_43),B_83))))) # label(fact_2014_add__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1198 (all Z1 all Z2 all W hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z1),Z2)),W) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z1),W)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z2),W))) # label(fact_1616_real__add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1199 (all B_113 all A_136 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_136)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_113)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_136),B_113)))))) # label(fact_1314_mult__pos__pos) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1200 (all B_95 all C_53 all A_116 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_116)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_95),C_53)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_95),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_116),C_53)))))) # label(fact_1872_add__strict__increasing2) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1201 (all A_89 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,one_one_complex),A_89) = A_89) # label(fact_2118_mult__1) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1202 (all X_2 all Y_2 (X_2 = Y_2 <-> hAPP_real_real(sqrt,Y_2) = hAPP_real_real(sqrt,X_2))) # label(fact_3311_real__sqrt__eq__iff) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1203 (all K_2 all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_2)) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K_2),M)),K_2) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(suc,zero_zero_nat)))))) # label(fact_3236_int__power__div__base) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1204 (all A_181 hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_181),A_181))) # label(fact_978_dvd__refl) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1205 (all Z_1 zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(uminus_uminus_int,Z_1)),Z_1)) # label(fact_3528_zadd__zminus__inverse2) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1206 (all D all A all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A),B))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D),A)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D),B)))) # label(fact_4008_coprime__mul__eq) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1207 (all X_1 all Y_1 all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_1),Z_1))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),Z_1)),Y_1))))) # label(fact_2635_div__prop1) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1208 (all X_2 (zero_zero_real = hAPP_real_real(sqrt,X_2) <-> X_2 = zero_zero_real)) # label(fact_3315_real__sqrt__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1209 (all N all X_2 all Diff all F ((all M_1 all X hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),X),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),X)))) & zero_zero_real != X_2 & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) & F = hAPP_n546711566l_real(Diff,zero_zero_nat) -> (exists T (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,X_2))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),N))) = hAPP_real_real(F,X_2) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_2))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(abs_abs_real,T))))))) # label(fact_4931_Maclaurin__all__lt__objl) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1210 (all N (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_int_real(real_int,N))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N)))) # label(fact_4438_real__of__int__gt__zero__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1211 (all X_2 all Y_2 all P_1 ((-hBOOL(P_1) -> hAPP_nat_int(semiri1621563631at_int,Y_2) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,P_1),X_2),Y_2))) & (hBOOL(P_1) -> hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,P_1),X_2),Y_2)) = hAPP_nat_int(semiri1621563631at_int,X_2)))) # label(fact_3254_int__if__cong) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1212 (all X_46 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_46),one_one_nat) = X_46) # label(fact_81_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1213 (all N_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(real_nat,N_1)))) # label(fact_3632_real__of__nat__ge__zero) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1214 (all A_207 (is_int(A_207) -> A_207 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_207),zero_zero_int))) # label(fact_630_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1215 (all B_125 all A_149 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_149)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_125)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_149),B_125)))))) # label(fact_1240_mult__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1216 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),zero_zero_int)) -> hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),A_3)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_2),one_one_int)),A_3))),one_one_int))) # label(fact_2292_neg__zmod__mult__2) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1217 (all Z_2 all W_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_2)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),W_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),Z_2)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(nat,W_1)),hAPP_int_nat(nat,Z_2)))))) # label(fact_2753_nat__le__eq__zle) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1218 (all W_1 (hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,W_1)))) <-> hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,W_1))))) # label(fact_3275_neg__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1219 (all X_2 all Y_2 ((hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_2),zero_zero_real)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_2))) & (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),zero_zero_real)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_2))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_2),Y_2))))) # label(fact_3359_real__0__le__divide__iff) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1220 (all X_2 hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),cos)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_2),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_2))),hAPP_real_real(cos,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_2))),hAPP_real_real(cos,X_2)))))) # label(fact_4798_DERIV__cos__realpow2) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1221 (all Ma all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(semiri151668891at_rat,Ma)),hAPP_nat_rat(semiri151668891at_rat,N))))) # label(fact_2932_of__nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1222 (all Lx_1 all Rx_1 all Ry_1 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_1),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_1),Ry_1)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_1),Rx_1)),Ry_1)) # label(fact_176_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1223 (all Y_2 all X_2 all A (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(log(A),X_2)),hAPP_real_real(log(A),Y_2))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Y_2))))))) # label(fact_3870_log__le__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1224 (all A_13 all B_10 all Y_1 all R_7 (is_int(B_10) & is_int(Y_1) -> (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_10),Y_1)),R_7) = A_13 -> ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_10)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_10),R_7)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_7),zero_zero_int))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_10)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_7)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_7),B_10))) -> (zero_zero_int != B_10 -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_13),B_10) = Y_1))))) # label(fact_2695_divmod__int__rel__div__eq) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1225 (all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hBOOL(hAPP_real_bool(isCont_real_real(hAPP_n546711566l_real(root,N)),zero_zero_real)))) # label(fact_5168_isCont__root__zero) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1226 (all M_7 all N_21 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(semiri984289939at_nat,M_7)),hAPP_nat_nat(semiri984289939at_nat,N_21)) = hAPP_nat_nat(semiri984289939at_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_7),N_21))) # label(fact_2967_of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1227 (all K_2 all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_2),K_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,I)),K_2))))) # label(fact_2864_less__trans__Suc) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1228 (all Y_2 all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_2),X_2)) -> X_2 = comple124823625p_real(hAPP_r1134773055l_bool(ord_at1589558736t_real(Y_2),X_2)))) # label(fact_5063_Sup__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1229 (all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(semiri132038758t_real,N))))) # label(fact_3148_of__nat__0__less__iff) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1230 (all R1 all A_3 all R2 all B_2 rcis(hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,R1),R2),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_3),B_2)) = hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,rcis(R1,A_3)),rcis(R2,B_2))) # label(fact_4264_rcis__divide) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1231 (all K_2 hAPP_int_rat(number_number_of_rat,K_2) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(number_number_of_int,K_2)),one_one_int)) # label(fact_4708_rat__number__expand_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1232 (all Lx all Rx all Ry hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Rx),Ry)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Rx),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx),Ry))) # label(fact_178_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1233 (all P (is_bool(P) -> fFalse = P | P = fTrue)) # label(help_fFalse_1_1_T) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1234 (all B_94 all C_52 all A_115 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_115)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_94),C_52)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_94),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_115),C_52)))))) # label(fact_1877_add__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1235 (all B_70 all A_81 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_81),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_70),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_81),B_70)),zero_zero_real))))) # label(fact_2187_add__neg__neg) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1236 (all X_1 all A_3 all B_2 hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_3),B_2)) = hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3)),B_2)) # label(fact_4381_powr__powr) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1237 (all M_25 all N_65 all A_216 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_216)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_216),M_25)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_216),N_65))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_25),N_65))))) # label(fact_530_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1238 (all A all X_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),archim856651990g_real(X_2))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,A)),X_2)))) # label(fact_4420_less__ceiling__eq) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1239 (all N_1 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (zero_zero_real = X_1 -> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1) = zero_zero_real) & (X_1 != zero_zero_real -> hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),hAPP_nat_real(real_nat,N_1)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1))))) # label(fact_4474_powr__realpow2) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1240 (all B_112 all C_67 all A_134 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_134)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_112),C_67)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_112),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_134),C_67)))))) # label(fact_1337_pos__add__strict) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1241 (all X_2 (hAPP_nat_nat(suc,zero_zero_nat) = hAPP_nat_nat(div_mod_nat(X_2),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) <-> -hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2)))) # label(fact_3774_odd__nat__equiv__def) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1242 (all P_1 all P1 all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D)) -> ((all X all K (is_int(K) & is_int(X) -> (hBOOL(hAPP_int_bool(P1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D)))) <-> hBOOL(hAPP_int_bool(P1,X))))) -> ((exists Z all X (is_int(X) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),Z)) -> (hBOOL(hAPP_int_bool(P_1,X)) <-> hBOOL(hAPP_int_bool(P1,X)))))) -> ((exists X1 hBOOL(hAPP_int_bool(P1,X1))) -> (exists X1 (is_int(X1) & hBOOL(hAPP_int_bool(P_1,X1))))))))) # label(fact_4006_minusinfinity) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1243 (all A_3 hAPP_int_int(uminus_uminus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(uminus_uminus_int,A_3)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) # label(fact_3549_power3__minus) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1244 (all A_225 all M_28 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_225),one_on1645066479umeral)),M_28) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_225),M_28)),M_28)) # label(fact_395_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1245 (all N_1 all M all A_3 all B_2 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2) = one_one_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_3),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_2),M)) = one_one_nat)) # label(fact_4300_coprime__exp2__nat) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1246 (all N_1 all M all R_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),R_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N_1),R_3)) -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N_1) = one_one_nat -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1)),R_3)))))) # label(fact_4366_divides__mult__nat) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1247 (all N_1 all I ((zero_z126310315umeral = I -> hAPP_nat_nat(code_nat_of_aux(I),N_1) = N_1) & (zero_z126310315umeral != I -> hAPP_nat_nat(code_nat_of_aux(I),N_1) = hAPP_nat_nat(code_nat_of_aux(hAPP_C498520661umeral(hAPP_C1594335432umeral(minus_1690775515umeral,I),one_on1645066479umeral)),hAPP_nat_nat(suc,N_1))))) # label(fact_3256_nat__of__aux__code) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1248 (all C_38 all A_95 all B_78 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_95),B_78)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_38),A_95)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_38),B_78))))) # label(fact_2054_add__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1249 (all A_91 (is_int(A_91) -> A_91 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_91),one_one_int))) # label(fact_2100_mult_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1250 (all V_2 ((hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> hAPP_nat_nat(suc,hAPP_int_nat(number_number_of_nat,V_2)) = one_one_nat) & (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> hAPP_nat_nat(suc,hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(succ,V_2))))) # label(fact_3304_Suc__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1251 (all L all F ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,N_2)),hAPP_nat_real(F,hAPP_nat_nat(suc,N_2))))) -> ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,N_2)),L))) -> ((all E (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),E)) -> (exists N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,L),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(F,N_2)),E)))))) -> hBOOL(tendsto_nat_real(F,L,sequentially)))))) # label(fact_5156_increasing__LIMSEQ) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1252 (all N_1 zero_zero_nat != hAPP_nat_nat(fact_fact_nat,N_1)) # label(fact_3800_fact__nonzero__nat) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1253 (all A_33 all B_29 all C_12 ((zero_z891286103de_int = C_12 -> zero_z891286103de_int = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_12),A_33)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_12),B_29))) & (zero_z891286103de_int != C_12 -> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_33),B_29) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_12),A_33)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_12),B_29))))) # label(fact_2568_div__mult__mult1__if) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1254 (all A all C all B (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A),B)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B),C))))) # label(fact_2035_add__le__cancel__right) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1255 (all X_7 all Q_5 hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_7),hAPP_nat_nat(suc,Q_5)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,X_7),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_7),Q_5))) # label(fact_3021_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1256 (all M_22 all A_214 all N_62 all B_159 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_214),N_62)),B_159)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_22),N_62)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_214),M_22)),B_159))))) # label(fact_567_power__le__dvd) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1257 (all X_9 all Q_7 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_9),Q_7)),X_9) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_9),hAPP_nat_nat(suc,Q_7))) # label(fact_3010_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1258 (all W all V_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_2))) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,W)),hAPP_int_int(number_number_of_int,V_2))) -> posDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(number_number_of_int,V_2)) = hAPP_P1975530577nt_int(adjust(hAPP_int_int(number_number_of_int,V_2)),posDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(number_number_of_int,V_2))))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,W)),hAPP_int_int(number_number_of_int,V_2))) -> posDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(number_number_of_int,V_2)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),hAPP_int_int(number_number_of_int,W))))) # label(fact_2503_posDivAlg__eqn__number__of) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1259 (all X_1 all D_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),D_3)) -> hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,X_1)),hAPP_nat_real(real_nat,D_3)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X_1),D_3))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,hAPP_nat_nat(div_mod_nat(X_1),D_3))),hAPP_nat_real(real_nat,D_3))))) # label(fact_3713_real__of__nat__div__aux) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1260 (all I all K_2 all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_2)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K_2)),J_2))) # label(fact_1679_diff__diff__right) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1261 (all Lx_6 all Ly_4 all Rx_6 all Ry_4 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_6),Ly_4)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_6),Ry_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_6),Rx_6)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ly_4),Ry_4))) # label(fact_138_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1262 (all B_114 all A_137 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_137)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_114),zero_zero_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_137),B_114)),zero_zero_nat))))) # label(fact_1309_mult__pos__neg) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1263 (all A_198 hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_198),one_one_nat) = A_198) # label(fact_747_power__one__right) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1264 (all X_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(exp_real,X_1)))) # label(fact_3832_exp__ge__zero) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1265 (all V all W_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,V)),hAPP_int_int(number_number_of_int,W_1))) <-> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,W_1)),hAPP_int_int(number_number_of_int,V))))) # label(fact_85_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1266 (all M all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,M)),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1))) # label(fact_3070_mult__Suc) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1267 (all P all Q all R hAPP_n215258509l_bool(P,hAPP_nat_nat(Q,R)) = hAPP_n215258509l_bool(hAPP_f66927821l_bool(hAPP_f2037783381l_bool(cOMBB_1146692694ol_nat,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HO) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1268 (all C_1 all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2)) -> (C_1 = B_2 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),C_1))))) # label(fact_1701_dvd_Oord__le__eq__trans) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1269 (all W_17 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,W_17)),hAPP_int_rat(number_number_of_rat,W_17)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_int_rat(number_number_of_rat,W_17)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_16_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1270 (all Lx_6 all Ly_4 all Rx_6 all Ry_4 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_6),Ly_4)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_6),Ry_4)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_6),Rx_6)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Ly_4),Ry_4))) # label(fact_142_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1271 (all A_184 hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_184),zero_zero_complex))) # label(fact_913_dvd__0__right) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1272 (all N_58 ((zero_zero_nat != N_58 -> zero_z891286103de_int = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,zero_z891286103de_int),N_58)) & (N_58 = zero_zero_nat -> one_on1684967323de_int = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,zero_z891286103de_int),N_58)))) # label(fact_599_power__0__left) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1273 (all N_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_1))))) # label(fact_3660_real__of__nat__Suc__gt__zero) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1274 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,K1)),hAPP_int_int(bit0,K2))))) # label(fact_127_less__eq__int__code_I15_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1275 (all N_1 all A_3 all B_2 all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),B_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N_1)),A_3)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N_1)),B_2)))))) # label(fact_4193_prime__divprod__pow) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1276 (all M_7 all N_21 hAPP_nat_rat(semiri151668891at_rat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_7),N_21)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(semiri151668891at_rat,M_7)),hAPP_nat_rat(semiri151668891at_rat,N_21))) # label(fact_2963_of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1277 (all A all B all C (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),A)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B)))))) # label(fact_1402_mult__le__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1278 (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2)) <-> hAPP_real_real(hAPP_r1250527377l_real(powr,X_2),one_one_real) = X_2)) # label(fact_4433_powr__one__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 5.39/5.44 1279 (all B_2 all Q_2 all R_3 all B_5 all Q_8 all R_5 (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_2)),R_3) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_5),Q_8)),R_5) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_5),Q_8)),R_5))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_5),B_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_5),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_2),Q_8))))))))) # label(fact_1496_zdiv__mono2__lemma) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1280 (all Z_10 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Z_10),hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Z_10),Z_10)) # label(fact_499_semiring__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1281 (all R_3 hAPP_real_real(norm_norm_real,R_3) = hAPP_real_real(abs_abs_real,R_3)) # label(fact_3381_real__norm__def) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1282 (all X_1 all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(fequal_real,X_1),Y_1)) | X_1 != Y_1)) # label(help_fequal_2_1_fequal_000tc__RealDef__Oreal_T) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1283 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hAPP_P1175774780nt_int(product_fst_int_int,negDivAlg(A_3,B_2)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)))) # label(fact_4224_div__neg__pos) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1284 (all A all B (zero_zero_rat = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),B) <-> zero_zero_rat = A | B = zero_zero_rat)) # label(fact_1035_mult__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1285 (all A all W_1 (zero_zero_nat != hAPP_int_nat(number_number_of_nat,W_1) & A = zero_zero_real <-> zero_zero_real = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A),hAPP_int_nat(number_number_of_nat,W_1)))) # label(fact_1384_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1286 (all X_2 all Y_2 all Xa_1 (of_real_complex(Xa_1) = hAPP_real_complex(hAPP_r265291036omplex(complex,X_2),Y_2) <-> X_2 = Xa_1 & Y_2 = zero_zero_real)) # label(fact_3986_complex__eq__cancel__iff2) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1287 (all B_2 all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_3)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,natfloor(A_3)),natfloor(B_2))),natfloor(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_3),B_2))))))) # label(fact_3402_le__mult__natfloor) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1288 (all N_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),one_one_nat)),N_1))) # label(fact_4028_coprime__plus1) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1289 (all M_26 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,M_26),M_26) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,one_on1645066479umeral),one_on1645066479umeral)),M_26)) # label(fact_409_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1290 (all Q_2 all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (hBOOL(hAPP_nat_bool(prime,Q_2)) -> (Q_2 != P_3 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,P_3),Q_2)))))) # label(fact_4165_distinct__prime__coprime) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1291 (all A_175 all E_3 all B_149 all C_87 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_175),E_3)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_149),E_3)),C_87)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_175),B_149)),E_3)),C_87)) # label(fact_1065_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1292 (all K_2 hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(number_number_of_int,K_2)),one_one_int) = hAPP_int_rat(number_number_of_rat,K_2)) # label(fact_4707_rat__number__collapse_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1293 (all X_2 all A_1 (hBOOL(hAPP_real_bool(A_1,X_2)) <-> hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),A_1)))) # label(fact_203_mem__def) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1294 (all X_1 all A_3 archim856651990g_real(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_int_real(real_int,A_3))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,archim856651990g_real(X_1)),A_3)) # label(fact_4426_ceiling__add) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1295 (all Z_1 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(hAPP_real_real(exp_real,hAPP_complex_real(re,Z_1)))),cis(hAPP_complex_real(im,Z_1))) = expi(Z_1)) # label(fact_4142_expi__def) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1296 (all P_1 P_1 = collec1979865426at_nat(P_1)) # label(fact_211_Collect__def) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1297 (all Y_21 all X_27 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_27)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_21)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_21),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_27),Y_21)),X_27)))))) # label(fact_1471_mult__right__le__one__le) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1298 (all D all Sa exists Z all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Z)) -> (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Sa))) <-> -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Sa)))))) # label(fact_2264_minf_I10_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1299 (all A all B (zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B) <-> B = A)) # label(fact_2091_right__minus__eq) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1300 (all A (is_int(A) -> (zero_zero_int = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) <-> zero_zero_int = A))) # label(fact_776_zero__eq__power2) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1301 (all X_3 (hBOOL(cauchy_complex(X_3)) -> hBOOL(cauchy_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,im),X_3))))) # label(fact_4756_Im_OCauchy) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1302 (all P_1 all A0 all A1 (hBOOL(hAPP_P1555980039t_bool(accp_P490777396at_nat(nat_gcd_rel),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,A0),A1))) -> ((all X all Y (hBOOL(hAPP_P1555980039t_bool(accp_P490777396at_nat(nat_gcd_rel),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,X),Y))) -> ((Y != zero_zero_nat -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Y),hAPP_nat_nat(div_mod_nat(X),Y)))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,X),Y))))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,A0),A1))))) # label(fact_4623_nat__gcd_Opinduct) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1303 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A),B))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),zero_zero_int))))) # label(fact_2627_neg__imp__zdiv__nonneg__iff) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1304 (all A_3 hAPP_real_real(cos,A_3) = hAPP_complex_real(re,cis(A_3))) # label(fact_4145_Re__cis) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1305 (all B_1_1 is_bool(summable_real(B_1_1))) # label(gsy_c_Series_Osummable_000tc__RealDef__Oreal) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1306 (all P_1 all N all K_1 (is_int(N) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N),K_1)),hAPP_int_int(div_mod_int(N),K_1))) <-> (all I_2 all J_1 (is_int(I_2) & is_int(J_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,J_1),K_1)) & hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),I_2)),J_1) = N & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),J_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,I_2),J_1))))))))) # label(fact_2701_split__pos__lemma) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1307 (all A_11 all N_18 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_11),N_18)),A_11) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_11),hAPP_nat_nat(suc,N_18))) # label(fact_3028_power__Suc2) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1308 (all C_8 all A_24 all B_21 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_24),B_21)) -> hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_21),C_8)),A_24) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,B_21),A_24)),C_8))) # label(fact_2601_dvd__div__mult) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1309 (all N_67 all A_218 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_218)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_218),N_67)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_218),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_218),N_67)))))) # label(fact_521_power__less__power__Suc) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1310 (all A all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),zero_zero_int),P_5)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_5),A)) -> hAPP_nat_int(semiri1621563631at_int,hAPP_f521865025ol_nat(finite1876863882t_bool,setS(A,P_5))) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3305_SetS__card) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1311 (all X_1 all A_3 zero_zero_real != hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3)) # label(fact_4377_powr__not__zero) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1312 (all W hAPP_int_rat(number_number_of_rat,W) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,W),one_one_int)) # label(fact_4680_rat__number__of__def) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1313 (all P_2 all P_1 all X_2 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) -> (hBOOL(P_2) <-> hBOOL(P_1))) -> ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) -> hBOOL(P_2)) <-> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) -> hBOOL(P_1))))) # label(fact_1750_imp__le__cong) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1314 (all V_12 all W_10 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_12)),hAPP_int_real(number267125858f_real,W_10)) = hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_12),W_10))) # label(fact_438_add__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1315 (all A_239 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_239),A_239) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_239),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_36_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1316 (all A_67 all C_27 all B_59 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(div_mod_int(A_67),C_27)),B_59)),C_27) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_67),B_59)),C_27)) # label(fact_2334_zmod__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1317 (all I all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(suc,I))),N_1)))) # label(fact_3137_diff__Suc__less) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1318 (all X_45 all P_7 all Q_11 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_45),P_7)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_45),Q_11)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_45),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_11))) # label(fact_99_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1319 (all C_33 all D_5 all A_78 all B_67 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_78),B_67)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_33),D_5)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_78),C_33)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_67),D_5)))))) # label(fact_2213_add__less__le__mono) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1320 (all Ma all N (-hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),N))) <-> -(-hBOOL(hAPP_nat_bool(even_odd_even_nat,N)) <-> -hBOOL(hAPP_nat_bool(even_odd_even_nat,Ma))))) # label(fact_3754_odd__add) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1321 (all A_3 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,one_one_nat),A_3))) # label(fact_4014_coprime__1_H) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1322 (all M all N_1 (M = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1) -> N_1 = zero_zero_nat)) # label(fact_1745_add__eq__self__zero) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1323 (all A_166 all B_141 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_166),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_141),A_166)))) # label(fact_1133_dvd__triv__right) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1324 (all F ((all I_2 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,hAPP_nat_nat(suc,I_2))),hAPP_nat_nat(F,I_2)))) -> (exists I_2 all K hAPP_nat_nat(F,I_2) = hAPP_nat_nat(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_2),K))))) # label(fact_4570_weak__decr__stable) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1325 (all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(sqrt,Y_2))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_2)))) # label(fact_3329_real__sqrt__gt__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1326 (all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,Y_1)),one_one_real)) -> hAPP_real_real(cos,hAPP_real_real(arccos,Y_1)) = Y_1)) # label(fact_3603_cos__arccos__abs) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1327 (all X_22 all Y_17 hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_22),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_17),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_22)),Y_17)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_22),Y_17)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_1620_power2__diff) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1328 (all Ma all N (hAPP_n1108039445umeral(semiri1619134803umeral,Ma) = hAPP_n1108039445umeral(semiri1619134803umeral,N) <-> Ma = N)) # label(fact_2831_of__nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1329 (all X_30 all N_34 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_34)) | one_one_rat = X_30 -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,X_30),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_30),N_34))))) # label(fact_996_dvd__power) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1330 (all M all N_1 (N_1 = M -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)))) # label(fact_1726_eq__imp__le) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1331 (all C all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C),A)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C),B))))) # label(fact_2067_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1332 (all N_32 all M_18 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),M_18)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N_32)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_18),N_32)))))) # label(fact_1363_less__1__mult) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1333 (all N_1 all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M)) -> hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_nat_real(real_nat,M)),hAPP_nat_real(real_nat,N_1)) = hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)))) # label(fact_3670_real__of__nat__diff) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1334 (all A_219 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_219),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_219),A_219)),A_219)) # label(fact_485_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1335 (all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),hAPP_int_int(bit0,K_1))))) # label(fact_275_rel__simps_I21_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1336 (all M_22 all A_214 all N_62 all B_159 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_214),N_62)),B_159)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_22),N_62)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_214),M_22)),B_159))))) # label(fact_573_power__le__dvd) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1337 (all A all B (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A)) | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B),zero_zero_rat)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A),zero_zero_rat)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),B))))) # label(fact_1250_zero__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1338 (all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,zero_zero_nat),N_1) = zero_zero_nat) # label(fact_1753_mult__0) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1339 (all A all B (hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A),B) = one_one_nat <-> (all D_2 (one_one_nat = D_2 <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),A)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),B)))))) # label(fact_4349_coprime__nat) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1340 (all A_3 all P_3 ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),A_3)) -> hAPP_int_int(legendre(A_3),P_3) = hAPP_int_int(number_number_of_int,min)) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),A_3)) -> hAPP_int_int(legendre(A_3),P_3) = one_one_int)) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),P_3)) -> hAPP_int_int(legendre(A_3),P_3) = zero_zero_int))) # label(fact_1626_Legendre__def) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1341 (all C_1 all N_1 all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),B_2)) -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,C_1),N_1) -> (exists R_1 exists S_2 (B_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,S_2),N_1) & A_3 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,R_1),N_1)))))) # label(fact_4479_coprime__pow) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1342 (all X_1 all M all Y_1 (hAPP_int_int(div_mod_int(Y_1),M) = hAPP_int_int(div_mod_int(X_1),M) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),Y_1),M)))) # label(fact_2401_Residues_Oaux) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1343 (all X_1 all Y_1 hAPP_int_int(hAPP_int_fun_int_int(int_gcd,X_1),Y_1) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(nat_gcd(hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,X_1))),hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,Y_1))))) # label(fact_4622_Nitpick_Oint__gcd__def) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1344 (all X_1 all Y_1 ratreal(hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,ratreal(X_1)),ratreal(Y_1))) # label(fact_4701_real__minus__code) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1345 (all C_83 all D_16 all A_169 all B_144 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_169),B_144)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,C_83),D_16)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_169),C_83)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_144),D_16)))))) # label(fact_1111_mult__dvd__mono) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1346 (all W hAPP_int_complex(ring_11397209091omplex,W) = hAPP_int_complex(number528085621omplex,W)) # label(fact_4582_complex__number__of__def) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1347 (all K_2 all L_2 hAPP_int_int(bit1,L_2) != hAPP_int_int(bit0,K_2)) # label(fact_351_rel__simps_I49_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1348 (all C_81 all A_167 all B_142 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_167),B_142)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_167),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_142),C_81))))) # label(fact_1126_dvd__mult2) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1349 (all N_1 all M M = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),M)),N_1)) # label(fact_1647_diff__add__inverse) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1350 (all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,N_1),zero_zero_nat) = N_1) # label(fact_5188_max__0R) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1351 (all A_3 all N_1 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_3),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_2),N_1))) -> (zero_zero_nat != N_1 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2))))) # label(fact_1591_divides__rev) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1352 (all M_19 all N_38 all A_188 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_188)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_188),M_19)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_188),N_38))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_19),N_38))))) # label(fact_884_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1353 (all B_143 all A_168 all C_82 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_168),C_82)) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_168),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_143),C_82))))) # label(fact_1115_dvd__mult) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1354 (all D_3 all A_3 all B_2 (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)) -> one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),A_3))) # label(fact_4999_coprime__lmult__int) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1355 (all A_3 cis(hAPP_real_real(uminus_uminus_real,A_3)) = hAPP_complex_complex(invers1449016382omplex,cis(A_3))) # label(fact_4149_cis__inverse) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1356 (all X_1 (is_int(X_1) -> (hBOOL(hAPP_int_bool(even_odd_even_int,X_1)) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = X_1))) # label(fact_3229_two__times__even__div__two) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1357 (all W_1 all X_2 (X_2 = hAPP_int_rat(number_number_of_rat,W_1) <-> X_2 = hAPP_int_rat(number_number_of_rat,W_1))) # label(fact_257_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1358 (all X_2 (is_int(X_2) -> (-hBOOL(hAPP_int_bool(even_odd_even_int,X_2)) <-> (exists Y (X_2 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Y)),one_one_int) & is_int(Y)))))) # label(fact_3242_odd__equiv__def) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1359 (all A_172 all B_147 all C_85 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_172),B_147)),C_85)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,B_147),C_85)))) # label(fact_1088_dvd__mult__right) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1360 (all N_66 all A_217 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_217)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_217),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_217),N_66)))))) # label(fact_523_power__gt1__lemma) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1361 (all X_2 all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Y_2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,X_2)),hAPP_int_int(number_number_of_int,Y_2))))) # label(fact_90_less__number__of) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1362 (all A_213 all N_61 all B_158 (is_int(B_158) & is_int(A_213) -> (hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_213),N_61) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_158),N_61) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_213)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_158)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_61)) -> B_158 = A_213)))))) # label(fact_574_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1363 (all B_2 all A_3 all C_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),C_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),C_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),A_3)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),B_2))))) # label(fact_1156_Euler_Oaux2) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1364 (all P all Q all R hAPP_nat_real(hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,P),Q),R) = hAPP_nat_real(P,hAPP_nat_nat(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1365 (all D int_ge_less_than(D) = collec1347809874nt_int(produc93441119t_bool(hAPP_f428220345t_bool(hAPP_f1878066172t_bool(cOMBS_1720984575t_bool,hAPP_f1315737299t_bool(hAPP_f538565589t_bool(cOMBB_1211207212ol_int,cOMBB_bool_bool_int),hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,D)))),ord_less_int)))) # label(fact_4861_int__ge__less__than__def) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1366 (all A all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X_2)),A)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),hAPP_nat_real(real_nat,A)))))) # label(fact_3688_natceiling__le__eq) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1367 (all A_3 all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),A_3)) -> (exists X exists Y hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_3),Y)),one_one_nat) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),X))))) # label(fact_4206_bezout__prime) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1368 (all A all P_5 all X (hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X),setS(A,P_5))) -> hBOOL(hAPP_f448129468l_bool(finite_finite_int,X)))) # label(fact_3357_SetS__elems__finite) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1369 (all X_2 (hAPP_real_real(cos,X_2) != zero_zero_real -> hBOOL(hAPP_real_bool(isCont_real_real(tan),X_2)))) # label(fact_5159_isCont__tan) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1370 (all B_1_1 is_int(the_int(B_1_1))) # label(gsy_c_HOL_OThe_000tc__Int__Oint) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1371 (all X_1 all Y_1 ((zero_zero_nat = Y_1 -> bezw(X_1,Y_1) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,one_one_int),zero_zero_int)) & (zero_zero_nat != Y_1 -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_P1175774780nt_int(product_snd_int_int,bezw(Y_1,hAPP_nat_nat(div_mod_nat(X_1),Y_1)))),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_P1175774780nt_int(product_fst_int_int,bezw(Y_1,hAPP_nat_nat(div_mod_nat(X_1),Y_1)))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_snd_int_int,bezw(Y_1,hAPP_nat_nat(div_mod_nat(X_1),Y_1)))),hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X_1),Y_1))))) = bezw(X_1,Y_1)))) # label(fact_4229_bezw_Osimps) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1372 (all B_2 all Q_8 all R_5 all Q_2 all R_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_8)),R_5)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_2)),R_3))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_3),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),R_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),R_5)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_2),Q_8))))))) # label(fact_1495_unique__quotient__lemma__neg) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1373 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),B)))) # label(fact_2214_le__iff__diff__le__0) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1374 (all M all K_2 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K_2)),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N_1)))) # label(fact_1775_add__leD2) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1375 (all C_37 all A_94 all B_77 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_94),B_77)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_94),C_37)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_77),C_37))))) # label(fact_2062_add__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1376 (all Ma all N (hAPP_nat_int(semiri1621563631at_int,N) = hAPP_nat_int(semiri1621563631at_int,Ma) <-> N = Ma)) # label(fact_3253_int__int__eq) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1377 (all Y_2 all X_2 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zOdd)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),Y_2)),zOdd))))) # label(fact_3099_odd__minus__even) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1378 (all Lx_5 all Ly_3 all Rx_5 all Ry_3 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_5),Ly_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_5),Ry_3)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_5),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_5),Ly_3)),Ry_3))) # label(fact_144_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1379 (all A_3 all B_2 all C_1 all N_1 (hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,C_1),N_1) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N_1)) -> (hAPP_int_int(legacy_zgcd(A_3),B_2) = one_one_int -> (exists K (is_int(K) & hAPP_int_int(abs_abs_int,A_3) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K),N_1))))))) # label(fact_4530_int__relprime__power__divisors) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1380 (all A_174 all B_148 all C_86 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_174),B_148)),C_86) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_174),C_86)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_148),C_86))) # label(fact_1071_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1381 (all X_1 all Y_1 hAPP_complex_complex(cnj,hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,X_1),Y_1)) = hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,hAPP_complex_complex(cnj,X_1)),hAPP_complex_complex(cnj,Y_1))) # label(fact_4078_complex__cnj__divide) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1382 (all W_1 all Y_2 all X_2 all Z_2 (hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,W_1),Y_2)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_2),Z_2)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,W_1),Z_2)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_2),Y_2)) <-> X_2 = W_1 | Y_2 = Z_2)) # label(fact_300_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1383 (all X_38 all Y_30 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_38),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_30),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_30)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_38),Y_30))))) # label(fact_785_power2__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1384 (all W_1 all X_2 (hAPP_int_complex(number528085621omplex,W_1) = X_2 <-> hAPP_int_complex(number528085621omplex,W_1) = X_2)) # label(fact_255_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1385 (all C_42 all A_99 all B_82 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_99),B_82)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_99),C_42)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_82),C_42))))) # label(fact_2023_add__right__mono) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1386 (all P all Q all R hAPP_nat_fun_nat_nat(hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,P),Q),R) = hAPP_nat_fun_nat_nat(P,hAPP_nat_nat(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1387 (all X_2 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,ratreal(X_2)),ratreal(Y_2))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_2),Y_2)))) # label(fact_4692_real__less__code) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1388 (all N all N_7 all F ((all N_2 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N_2)),hAPP_nat_nat(F,hAPP_nat_nat(suc,N_2))))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),N_7)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N)),hAPP_nat_nat(F,N_7)))))) # label(fact_3252_dvd_Olift__Suc__mono__le) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1389 (all N_1 all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(fact_fact_nat,M)),hAPP_nat_nat(fact_fact_nat,N_1)))))) # label(fact_3805_fact__less__mono__nat) # label(axiom) # label(non_clause). [assumption]. 5.39/5.45 1390 (all N all Diff all F all H ((all M_1 all T (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),H)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),T)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N)) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),T),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T))))) & F = hAPP_n546711566l_real(Diff,zero_zero_nat) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H)) -> (exists T (hAPP_real_real(F,H) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,H))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),N))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),H)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),T)))))) # label(fact_4934_Maclaurin2__objl) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1391 (all X_2 all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,X_2)),hAPP_int_real(number267125858f_real,Y_2))))) # label(fact_91_less__number__of) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1392 (all A (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A),zero_zero_rat)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),A)),zero_zero_rat)))) # label(fact_2180_double__add__le__zero__iff__single__add__le__zero) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1393 (all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(fact_fact_int,X_1))))) # label(fact_4278_transfer__nat__int__factorial__closure) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1394 (all X_1 all Y_1 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(standardRes(M),X_1)),hAPP_int_int(standardRes(M),Y_1))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_1)),M)))) # label(fact_2294_StandardRes__prop4) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1395 (all A all B all P_1 ((all A_2 all B_4 all C_3 (hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),B_4))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),B_4)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_4),C_3)) & hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,B_4),C_3))) -> hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),C_3))))) -> ((all X exists D_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2)) & (all A_2 all B_4 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),X)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),B_4)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_4),A_2)),D_2)) -> hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),B_4))))))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) -> hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A),B))))))) # label(fact_4295_lemma__BOLZANO) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1396 (all X_17 all Y_13 all Ya_2 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_17),Y_13)),Ya_2) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_17),Ya_2)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Y_13),Ya_2))) # label(fact_2485_mult__left_Oadd) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1397 (all A_235 all B_164 all C_95 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_235),B_164)),C_95) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_235),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_164),C_95))) # label(fact_222_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1398 (all A_61 all B_53 all C_24 hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_61),B_53)),C_24) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_61),C_24)),B_53)),C_24)) # label(fact_2357_mod__add__left__eq) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1399 (all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arcsin,Y_1)))))) # label(fact_3506_arcsin__lbound) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1400 (all X_1 all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(exp_real,X_1)),hAPP_real_real(exp_real,Y_1))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1)))) # label(fact_3823_exp__less__cancel) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1401 (all N_1 all X_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N_1),X_1))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,N_1)),hAPP_nat_real(real_nat,X_1))))) # label(fact_3671_real__of__nat__div4) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1402 (all N_4 (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_4)) <-> (exists M_1 all X (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),N_4)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),M_1)))))) # label(fact_4624_finite__nat__set__iff__bounded__le) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1403 (all M all K_2 all N_1 (hAPP_int_int(legacy_zgcd(K_2),N_1) = one_one_int -> hAPP_int_int(legacy_zgcd(M),N_1) = hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),N_1))) # label(fact_4500_zgcd__zmult__cancel) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1404 (all A_213 all N_61 all B_158 (hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_158),N_61) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_213),N_61) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_213)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_158)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_61)) -> A_213 = B_158))))) # label(fact_579_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1405 (all X_1 hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(suc,X_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,X_1)),hAPP_nat_nat(fact_fact_nat,X_1))) # label(fact_3807_Fact_Ofact__Suc) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1406 (all P all Q all R hAPP_int_bool(P,hAPP_int_int(Q,R)) = hAPP_int_bool(hAPP_f429935748t_bool(hAPP_f1512942609t_bool(cOMBB_int_bool_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__HOL__Obool_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1407 (all A_3 all B_2 cis(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_3),B_2)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,cis(A_3)),cis(B_2))) # label(fact_4147_cis__mult) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1408 (all X0 all F all R_4 ((all X (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X),hAPP_r1134773055l_bool(ord_gr788844697n_real(hAPP_real_real(uminus_uminus_real,R_4)),R_4))) -> hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),F)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),suc)))),hAPP_r474017924t_real(power_power_real,X)))))) -> (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X0),hAPP_r1134773055l_bool(ord_gr788844697n_real(hAPP_real_real(uminus_uminus_real,R_4)),R_4))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_4)) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_f1122023986l_real(hAPP_f1786887945l_real(cOMBB_1896127834l_real,suminf_real),hAPP_f1993765077t_real(hAPP_f895854793t_real(cOMBB_1948233853l_real,hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),F))),hAPP_f1501741374t_real(hAPP_f1135648319t_real(cOMBC_1329014627t_real,hAPP_f1163894827t_real(hAPP_f1998755145t_real(cOMBB_421464275l_real,cOMBB_nat_real_nat),power_power_real)),suc))),X0),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),F)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),suc)))),hAPP_r474017924t_real(power_power_real,X0))))))))) # label(fact_4841_DERIV__power__series_H) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1409 (all C_65 all D_12 all A_131 all B_110 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_131),B_110)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_65),D_12)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_131)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_65)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_131),C_65)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_110),D_12)))))))) # label(fact_1417_mult__strict__mono_H) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1410 (all P_1 collec50511176nt_int(P_1) = P_1) # label(fact_208_Collect__def) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1411 (all A_9 all N_15 all B_8 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_9),hAPP_nat_nat(suc,N_15))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_8),hAPP_nat_nat(suc,N_15)))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_8)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_9),B_8))))) # label(fact_3110_power__le__imp__le__base) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1412 (all P all Q all R hAPP_f1591894897l_real(P,hAPP_r1250527377l_real(Q,R)) = hAPP_r579619957l_real(hAPP_f823702159l_real(hAPP_f1040184293l_real(cOMBB_557258983l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc__036) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1413 (all A_9 all N_15 all B_8 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_9),hAPP_nat_nat(suc,N_15))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_8),hAPP_nat_nat(suc,N_15)))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_8)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_9),B_8))))) # label(fact_3111_power__le__imp__le__base) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1414 (all C_39 all D_7 all A_96 all B_79 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_96),B_79)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,C_39),D_7)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_96),C_39)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_79),D_7)))))) # label(fact_2048_add__strict__mono) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1415 (all A all C all B (is_int(C) -> (C = zero_zero_int | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),B)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),C)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),C)))))) # label(fact_1381_dvd__mult__cancel__right) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1416 (all N_1 all M hAPP_Q1762011733de_int(div_mo231679042de_int(N_1),M) = hAPP_i1732201573de_int(quickcheck_of_int,hAPP_int_int(div_mod_int(quickcheck_int_of(N_1)),quickcheck_int_of(M)))) # label(fact_4552_mod__code__int__def) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1417 (all Ma all N (hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),N) = zero_zero_nat <-> N = zero_zero_nat | Ma = zero_zero_nat)) # label(fact_1755_mult__is__0) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1418 (all X_9 all Q_7 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_9),hAPP_nat_nat(suc,Q_7)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_9),Q_7)),X_9)) # label(fact_3008_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1419 (all A_3 all B_2 hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(uminus_uminus_int,A_3)),B_2) = hAPP_rat_rat(uminus_uminus_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2))) # label(fact_4688_minus__rat) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1420 (all A (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),A)),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),zero_zero_real)))) # label(fact_2179_double__add__le__zero__iff__single__add__le__zero) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1421 (all B_2 all A_3 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),B_2)) -> (exists D_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2)) & (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),Y))),D_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),Y)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),B_2))))))))) # label(fact_3412_lemma__interval__lt) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1422 (all A_163 hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,one_on1645066479umeral),A_163))) # label(fact_1149_one__dvd) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1423 (all A_88 A_88 = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,one_one_rat),A_88)) # label(fact_2127_mult__1__left) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1424 (all P all Q all R hAPP_P1982799375l_real(hAPP_r931665397l_real(P,R),hAPP_r1195171167l_real(Q,R)) = hAPP_r938550413l_real(hAPP_f1293550189l_real(hAPP_f183342685l_real(cOMBS_794971624l_real,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__prod_Itc__RealDef__Oreal_Mtc___037) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1425 (all V_2 all W hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_2),W)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,W))) # label(fact_366_plus__numeral__code_I9_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1426 (all Y_2 all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,X_2)),hAPP_real_real(ln,Y_2))))))) # label(fact_3437_ln__le__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1427 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sqrt,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_3343_real__sqrt__pow2__gt__zero) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1428 (all Y_1 all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1)) -> hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Y_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(nat,X_1)),hAPP_int_nat(nat,Y_1))))) # label(fact_2756_Nat__Transfer_Otransfer__nat__int__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1429 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(suc,N_1))))) # label(fact_2873_le__SucI) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1430 (all A_108 all B_87 all C_47 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_108),B_87)),C_47) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_108),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_87),C_47))) # label(fact_1950_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1431 (all X_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) <-> hBOOL(hAPP_int_bool(nat_is_nat,X_2)))) # label(fact_5128_is__nat__def) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1432 (all A_3 all B_2 all C_1 all D_3 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),B_2)),hAPP_real_complex(hAPP_r265291036omplex(complex,C_1),D_3)) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_3),C_1)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_2),D_3))) # label(fact_3961_complex__add) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1433 (all A_128 all C_62 all B_107 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_128),C_62)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_107),C_62))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_62)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_128),B_107))))) # label(fact_1433_mult__right__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1434 (all C all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),A)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C),zero_zero_real)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B),A)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)))) # label(fact_1324_mult__less__cancel__left__disj) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1435 (all A_233 all C_93 all D_18 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_93),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_233),D_18)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_233),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_93),D_18))) # label(fact_237_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1436 (all A_106 A_106 = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_106),zero_zero_real)) # label(fact_1973_add__0__right) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1437 (all V_13 all W_11 all Z_13 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_13)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,W_11)),Z_13)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_13),W_11))),Z_13)) # label(fact_435_add__number__of__left) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1438 (all P_3 all Y_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),Y_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_1),N_1)))))) # label(fact_934_zpower__zdvd__prop1) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1439 (all K_2 (is_int(K_2) -> K_2 = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),pls))) # label(fact_1532_diff__bin__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1440 (all N_1 hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),one_one_nat)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),one_one_nat)),hAPP_nat_nat(fact_fact_nat,N_1))) # label(fact_3812_fact__plus__one__nat) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1441 (all P all Q all R hAPP_f800510211t_bool(hAPP_i1796122964t_bool(P,R),Q) = hAPP_i418383825t_bool(hAPP_f289318786t_bool(hAPP_f729588221t_bool(cOMBC_678732631t_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1442 (all A (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),A))))) # label(fact_2181_zero__le__double__add__iff__zero__le__single__add) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1443 (all X_31 all Y_24 (Y_24 != X_31 -> (-hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_31),Y_24)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_24),X_31))))) # label(fact_971_linorder__neqE__linordered__idom) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1444 (all A_226 all B_161 all V_18 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_226),hAPP_int_real(number267125858f_real,V_18))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_161),hAPP_int_real(number267125858f_real,V_18))) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_226),B_161)),hAPP_int_real(number267125858f_real,V_18))) # label(fact_381_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1445 (all N_1 one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),one_one_nat)),N_1)) # label(fact_4350_coprime__plus__one__nat) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1446 (all X_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_2),one_one_int)) <-> hAPP_int_int(abs_abs_int,X_2) = one_one_int)) # label(fact_3424_zdvd1__eq) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1447 (all A_194 all B_153 all N_46 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_194),N_46)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_153),N_46)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_194),B_153)),N_46)) # label(fact_814_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1448 (all X_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X_2)),one_one_nat)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),one_one_real)))) # label(fact_3400_natceiling__le__eq__one) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1449 (all A_193 all M_21 all N_45 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_193),M_21)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_193),N_45)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_193),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_21),N_45))) # label(fact_820_power__add) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1450 (all X_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),pls)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,X_2)),zero_zero_real)))) # label(fact_760_less__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1451 (all A_232 all C_92 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_92),A_232) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_232),C_92)) # label(fact_242_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1452 (all M hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,M)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,M))) # label(fact_2849_int__Suc) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1453 (all A_221 (is_int(A_221) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,pls))),A_221) = A_221)) # label(fact_452_mult__numeral__1) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1454 (all M all N_1 hAPP_nat_nat(div_mod_nat(M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),N_1)) = hAPP_nat_nat(div_mod_nat(M),hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,N_1))))) # label(fact_3211_mod__Suc__eq__mod__add3) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1455 (all M all K_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),K_2)) -> hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M))),K_2) = one_one_nat)) # label(fact_3187_Suc__times__mod__eq) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1456 (all N_1 hAPP_int_nat(number_number_of_nat,N_1) = natceiling(hAPP_int_real(number267125858f_real,N_1))) # label(fact_3391_natceiling__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1457 (all P all Q all R hAPP_r712953929l_real(hAPP_f1108539711l_real(hAPP_f1109256153l_real(cOMBB_44084907l_real,P),Q),R) = hAPP_f1730463541l_real(P,hAPP_r579619957l_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__fun_Itc__RealDef__Oreal_044) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1458 (all N hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_eq_int),N)))))) # label(fact_4735_bdd__int__set__l__le__finite) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1459 (all A_3 all B_2 all Qr_1 (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2)),Qr_1)) -> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_P1975530577nt_int(negateSnd,Qr_1))))) # label(fact_4189_divmod__int__rel__neg) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1460 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M)) -> hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)),N_1))),hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)),N_1)) = divmod_nat(M,N_1)))) # label(fact_4205_divmod__nat__step) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1461 (all C_80 all A_164 all B_139 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_164),B_139)) -> (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_164),C_80)) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_164),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_139),C_80)))))) # label(fact_1143_dvd__add) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1462 (all B_36 zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,zero_zero_real),B_36)) # label(fact_2470_mult_Ozero__left) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1463 (all A all E_2 all C all B all D (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),E_2)),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),E_2)),D))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B)),E_2)),C)),D)))) # label(fact_1580_less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1464 (all B_1_1 all B_2_1 is_bool(hAPP_f526364711l_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__fun_Itc__prod_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__p) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1465 (all M_22 all A_214 all N_62 all B_159 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_214),N_62)),B_159)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_22),N_62)) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_214),M_22)),B_159))))) # label(fact_571_power__le__dvd) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1466 (all K_2 all M all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1))) # label(fact_1784_add__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1467 (all M all N_1 hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N_1))) # label(fact_3042_zadd__int) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1468 (all X_2 all Y_2 (is_int(X_2) & is_int(Y_2) -> (hAPP_int_complex(number528085621omplex,X_2) = hAPP_int_complex(number528085621omplex,Y_2) <-> Y_2 = X_2))) # label(fact_249_eq__number__of) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1469 (all N_1 all K_2 (is_int(K_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2)) -> hAPP_int_int(legacy_zgcd(K_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N_1)) = K_2))) # label(fact_4508_zgcd__zmult__eq__self) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1470 (all B_2 all A_3 all C_1 hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,B_2),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),C_1)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,B_2),C_1))) # label(fact_4980_gcd__int_Oleft__commute) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1471 (all Ta all D all D_1 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),D_1)) -> (all X all K (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X),Ta))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,K),D_1))),Ta))))))) # label(fact_1815_inf__period_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1472 (all D_1 all X_2 (X_2 != zero_zero_real -> ((hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2)) -> hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,X_2))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = D_1) -> ((hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),zero_zero_real)) -> D_1 = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,X_2)))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) -> hBOOL(hAPP_real_bool(deriv_real(sqrt,X_2),D_1)))))) # label(fact_3879_DERIV__real__sqrt__generic) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1473 (all N_1 all M (is_int(M) & is_int(N_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),N_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N_1),M)) -> M = N_1)))))) # label(fact_581_zdvd__antisym__nonneg) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1474 (all J_2 all K_2 all A_3 all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,J_2),K_2)),A_3),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(multInv(P_3),J_2)),J_2)),K_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(multInv(P_3),J_2)),A_3)),P_3)))) # label(fact_1894_aux______3) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1475 (all A_202 all N_52 all N_51 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_52),N_51)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_202)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_202),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_202),N_51)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_202),N_52))))))) # label(fact_677_power__decreasing) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1476 (all K_2 all L_2 hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),hAPP_int_int(succ,L_2))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit1,K_2)),hAPP_int_int(bit1,L_2))) # label(fact_3301_add__Bit1__Bit1) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1477 (all A_223 A_223 = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,pls)),A_223)) # label(fact_414_add__numeral__0) # label(axiom) # label(non_clause). [assumption]. 5.43/5.45 1478 (all N_32 all M_18 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),M_18)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),N_32)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,M_18),N_32)))))) # label(fact_1366_less__1__mult) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1479 (all C_34 all D_6 all A_79 all B_68 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_79),B_68)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_34),D_6)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_79),C_34)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_68),D_6)))))) # label(fact_2204_add__le__less__mono) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1480 (all N_25 hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),hAPP_n1108039445umeral(semiri1619134803umeral,N_25)))) # label(fact_2908_of__nat__0__le__iff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1481 (all B all A all Ma (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(bnorRset(A),Ma))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A)))) # label(fact_4292_Bnor__mem__zle) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1482 (all I_1 all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,I_1),K_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),I_1))))) # label(fact_2632_pos__imp__zdiv__pos__iff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1483 (all P all Q all R hAPP_f11151005nt_int(P,hAPP_P2110489235nt_int(Q,R)) = hAPP_P752050971nt_int(hAPP_f301246617nt_int(hAPP_f1151807883nt_int(cOMBB_563965291nt_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc___039) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1484 (all A_3 all N_1 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N_1))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),A_3))))) # label(fact_698_zprime__zdvd__power) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1485 (all P all Q all R hAPP_r1195171167l_real(P,hAPP_real_real(Q,R)) = hAPP_r1195171167l_real(hAPP_f617735571l_real(hAPP_f800361423l_real(cOMBB_2107848501l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__prod_Itc__RealDef__Oreal_Mtc__) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1486 (all P all Q all R hAPP_nat_bool(hAPP_f737665877t_bool(hAPP_f605995867t_bool(cOMBB_real_bool_nat,P),Q),R) = hAPP_real_bool(P,hAPP_nat_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__HOL__Obool_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1487 (all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat,P_3)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_2773_Int2_Oaux__1) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1488 (all A_232 all C_92 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_232),C_92) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_92),A_232)) # label(fact_241_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1489 (all A_88 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,one_on1645066479umeral),A_88) = A_88) # label(fact_2124_mult__1__left) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1490 (all X_1 hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,X_1))) = hAPP_int_int(abs_abs_int,X_1)) # label(fact_3425_int__nat__abs) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1491 (all B_49 all B_48 all A_57 all C_20 all A_56 (hAPP_Q1762011733de_int(div_mo231679042de_int(A_57),C_20) = hAPP_Q1762011733de_int(div_mo231679042de_int(A_56),C_20) -> (hAPP_Q1762011733de_int(div_mo231679042de_int(B_49),C_20) = hAPP_Q1762011733de_int(div_mo231679042de_int(B_48),C_20) -> hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_57),B_49)),C_20) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_56),B_48)),C_20)))) # label(fact_2373_mod__add__cong) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1492 (all K_2 all L_2 hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),L_2)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(bit0,K_2)),hAPP_int_int(bit0,L_2))) # label(fact_1533_diff__bin__simps_I7_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1493 (all Q_2 all R_3 all B_2 all C_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,R_3),B_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),hAPP_nat_nat(div_mod_nat(Q_2),C_1))),R_3)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),C_1)))))) # label(fact_2523_mod__lemma) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1494 (all X_2 all Y_2 all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),B)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B),X_2)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B),Y_2))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),Y_2))))) # label(fact_536_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1495 (all X_1 all Y_1 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(fequal_nat,X_1),Y_1)) | X_1 = Y_1)) # label(help_fequal_1_1_fequal_000tc__Nat__Onat_T) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1496 (all M_6 all N_20 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_nat_real(semiri132038758t_real,M_6)),N_20) = hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_6),N_20))) # label(fact_2984_of__nat__power) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1497 (all X_2 all W_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_2),hAPP_int_nat(number_number_of_nat,W_1)))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),X_2)) | zero_zero_nat = hAPP_int_nat(number_number_of_nat,W_1))) # label(fact_923_zero__less__power__nat__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1498 (all Diff all N all X_2 (X_2 = zero_zero_real -> (zero_zero_nat != N -> hAPP_real_real(hAPP_n546711566l_real(Diff,zero_zero_nat),zero_zero_real) = hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,X_2))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))))) # label(fact_4907_Maclaurin__zero) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1499 (all A all E_2 all C all B all D (D = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B)),E_2)),C) <-> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),E_2)),C) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),E_2)),D))) # label(fact_1552_eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1500 (all B_118 all A_141 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_141),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_118)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_141),B_118)),zero_zero_real))))) # label(fact_1283_mult__neg__pos) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1501 (all A_36 hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_36),zero_zero_nat) = zero_zero_nat) # label(fact_2538_div__by__0) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1502 (all B all A all F all G all X_2 all D_1 (hBOOL(hAPP_real_bool(deriv_real(F,hAPP_real_real(G,X_2)),D_1)) -> (D_1 != zero_zero_real -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),X_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),B)) -> ((all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),Y)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),B)) -> Y = hAPP_real_real(F,hAPP_real_real(G,Y)))) -> (hBOOL(hAPP_real_bool(isCont_real_real(G),X_2)) -> hBOOL(hAPP_real_bool(deriv_real(G,X_2),hAPP_real_real(inverse_inverse_real,D_1)))))))))) # label(fact_5174_DERIV__inverse__function) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1503 (all Lx_1 all Rx_1 all Ry_1 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_1),Rx_1)),Ry_1) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_1),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_1),Ry_1))) # label(fact_177_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1504 (all X_1 all Y_1 (hAPP_complex_real(re,X_1) = hAPP_complex_real(re,Y_1) -> (hAPP_complex_real(im,X_1) = hAPP_complex_real(im,Y_1) -> Y_1 = X_1))) # label(fact_4116_complex__eqI) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1505 (all K_2 all A_3 all B_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),K_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),K_2)),M)))) # label(fact_1162_zcong__scalar) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1506 (all N_1 -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N_1)),zero_zero_real))) # label(fact_3646_not__real__of__nat__less__zero) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1507 (all A_121 all C_56 all B_101 all D_9 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_121),B_101)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,C_56),D_9)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_121),C_56)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_101),D_9))) # label(fact_1527_add__diff__add) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1508 (all B_150 all A_178 (zero_zero_rat != A_178 -> (zero_zero_rat != B_150 -> hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_178),B_150) != zero_zero_rat))) # label(fact_1031_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1509 (all N_12 all R_6 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_6)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_6),one_one_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,R_6),hAPP_nat_nat(suc,N_12))),R_6))))) # label(fact_3161_realpow__Suc__le__self) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1510 (all V_17 all B_160 all C_90 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_17)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_160),C_90)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_17)),B_160)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_17)),C_90))) # label(fact_388_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1511 (all X_2 all N (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,N))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),hAPP_int_real(number267125858f_real,N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X_2)),hAPP_int_nat(number_number_of_nat,N))))))) # label(fact_3410_natceiling__le__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1512 (all A_3 hAPP_real_real(hAPP_r1250527377l_real(powr,one_one_real),A_3) = one_one_real) # label(fact_4378_powr__one__eq__one) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1513 (all B_69 all A_80 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_80)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_69)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_80),B_69)))))) # label(fact_2190_add__pos__pos) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1514 (all Ma all N (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N)),Ma)))) # label(fact_2876_not__less__eq__eq) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1515 (all A_231 all N_68 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_231),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_68)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_231),N_68)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_283_power__even__eq) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1516 (all X_1 all Y_1 (hBOOL(hAPP_int_bool(even_odd_even_int,Y_1)) -> hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_1))))) # label(fact_2886_anything__times__even) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1517 (all A_97 all C_40 all B_80 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_97),C_40)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_80),C_40))) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_97),B_80)))) # label(fact_2046_add__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1518 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),zero_zero_int)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),zero_zero_int)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),B)),zero_zero_int)))) # label(fact_1245_mult__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1519 (all Lx_1 all Rx_1 all Ry_1 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_1),Ry_1)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_1),Rx_1)),Ry_1)) # label(fact_172_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1520 (all A all B (is_int(B) & is_int(A) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),zero_zero_int)) <-> B = A))) # label(fact_1157_IntPrimes_Ozcong__zero) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1521 (all X_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_complex_real(im,X_1))),hAPP_complex_real(norm_norm_complex,X_1)))) # label(fact_4102_abs__Im__le__cmod) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1522 (all W ii != hAPP_int_complex(number528085621omplex,W)) # label(fact_3976_complex__i__not__number__of) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1523 (all A_215 all M_24 all N_64 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_24),N_64)) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_215),M_24)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_215),N_64))))) # label(fact_557_le__imp__power__dvd) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1524 (all A_104 A_104 = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,zero_zero_complex),A_104)) # label(fact_1991_add__0__left) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1525 (all A (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),A))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A)))) # label(fact_2183_zero__le__double__add__iff__zero__le__single__add) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1526 (all N_1 (hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),one_one_nat)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3787_lemma__even__div2) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1527 (all A_3 all Y_1 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1)) -> hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_1),Y_1)),A_3) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,Y_1),A_3))))) # label(fact_4434_powr__divide) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1528 (all A_197 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_197),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_782_zero__le__power2) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1529 (all Lx_6 all Ly_4 all Rx_6 all Ry_4 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_6),Ly_4)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_6),Ry_4)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_6),Rx_6)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Ly_4),Ry_4))) # label(fact_141_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1530 (all A_3 all N_1 all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_3),N_1))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),A_3))))) # label(fact_4174_prime__dvd__power) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1531 (all A (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) <-> A != zero_zero_real)) # label(fact_799_zero__less__power2) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1532 (all X_1 all Y_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),Y_1)))) # label(fact_4402_powr__ge__pzero) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1533 (all A_12 all B_9 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_12),B_9)),B_9)),hAPP_Q1762011733de_int(div_mo231679042de_int(A_12),B_9)) = A_12) # label(fact_2711_mod__div__decomp) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1534 (all X_36 all Y_28 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_36),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_28),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_28)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_36),Y_28))))) # label(fact_839_power2__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1535 (all P_1 all A0 (hBOOL(hAPP_int_bool(accp_int(quickc1265749348ro_rel),A0)) -> ((all I_2 (is_int(I_2) -> (hBOOL(hAPP_int_bool(accp_int(quickc1265749348ro_rel),I_2)) -> ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_2),zero_zero_int)) -> (zero_zero_int != I_2 -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,I_2),one_one_int))))) -> hBOOL(hAPP_int_bool(P_1,I_2)))))) -> hBOOL(hAPP_int_bool(P_1,A0))))) # label(fact_4269_around__zero_Opinduct) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1536 (all Y_6 all P_6 all N_10 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_6),N_10)) -> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N_10)),P_6)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),P_6))),Y_6))) # label(fact_3180_lemma__realpow__diff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1537 (all X_1 (hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,X_1)),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X_1),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))))) # label(fact_3771_even__nat__plus__one__div__two) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1538 (all M_25 all N_65 all A_216 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_216)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_216),M_25)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_216),N_65))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_25),N_65))))) # label(fact_532_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1539 (all N_1 all X_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,N_1)),hAPP_int_real(real_int,X_1))),hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N_1),X_1)))),one_one_real))) # label(fact_4464_real__of__int__div3) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1540 (all Z_12 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_12) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z_12),Z_12)) # label(fact_489_semiring__mult__2) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1541 (all C_1 all D_3 all A_3 all B_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C_1),D_3),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),C_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_2),D_3)),M))))) # label(fact_1166_zcong__zadd) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1542 (all Ma all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)) & N != Ma <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)))) # label(fact_1771_nat__less__le) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1543 (all A_21 all C_6 all B_18 (is_int(B_18) -> (B_18 != zero_zero_int -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_21),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_6),B_18))),B_18) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_6),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_21),B_18))))) # label(fact_2647_div__mult__self1) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1544 (all X_22 all Y_17 hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_22),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_17),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_22)),Y_17)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_22),Y_17)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_1621_power2__diff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1545 (all G all X_2 all Ma (hBOOL(hAPP_real_bool(deriv_real(G,X_2),Ma)) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,sin),G),X_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,hAPP_real_real(G,X_2))),Ma))))) # label(fact_4761_DERIV__fun__sin) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1546 (all V_12 all W_10 hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_12),W_10)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_12)),hAPP_int_complex(number528085621omplex,W_10))) # label(fact_439_add__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1547 (all Ma all N (Ma != zero_zero_nat | zero_zero_nat != N <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Ma),N))))) # label(fact_4343_gcd__pos__nat) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1548 (all A_74 hAPP_nat_nat(div_mod_nat(A_74),zero_zero_nat) = A_74) # label(fact_2307_mod__by__0) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1549 (all X_2 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,X_2)),hAPP_int_real(real_int,Y_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),Y_2)))) # label(fact_4386_real__of__int__le__iff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1550 (all N_63 all M_23 all X_42 all Y_34 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,X_42),Y_34)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_63),M_23)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_42),N_63)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_34),M_23)))))) # label(fact_560_dvd__power__le) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1551 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M)),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)))) # label(fact_3060_Suc__le__lessD) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1552 (all A_232 all C_92 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_92),A_232) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_232),C_92)) # label(fact_245_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1553 (all N_44 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,one_one_nat),N_44) = one_one_nat) # label(fact_826_power__one) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1554 (all V_15 all W_13 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_15)),hAPP_int_int(number_number_of_int,W_13)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_15),W_13))) # label(fact_425_arith__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1555 (all B_2 (is_int(B_2) -> (B_2 != zero_zero_int -> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(zero_zero_int,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),zero_zero_int)))))) # label(fact_2792_divmod__int__rel__0) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1556 (all N_1 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),N_1) = hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,pi),hAPP_nat_real(real_nat,N_1)))) # label(fact_3715_cos__npi2) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1557 (all K_1 all F all N all Ma (is_int(K_1) -> ((all I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),N)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),I_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_2),one_one_nat))),hAPP_nat_int(F,I_2)))),one_one_int)))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,Ma)),K_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),hAPP_nat_int(F,N))) -> (exists I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),N)) & K_1 = hAPP_nat_int(F,I_2) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),I_2)))))))))) # label(fact_3504_nat__intermed__int__val) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1558 (all A_46 all B_39 all C_14 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_46),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_39),C_14))),B_39) = hAPP_int_int(div_mod_int(A_46),B_39)) # label(fact_2419_mod__mult__self2) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1559 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,X_1)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),one_one_real))))) # label(fact_3445_ln__le__minus__one) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1560 (all M all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N_1))) # label(fact_2879_diff__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1561 (all X_2 all Y_2 (zero_zero_rat = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) <-> Y_2 = zero_zero_rat & zero_zero_rat = X_2)) # label(fact_803_sum__power2__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1562 (all X_2 (one_one_complex = X_2 <-> X_2 = one_one_complex)) # label(fact_1956_one__reorient) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1563 (all A all R_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_2)) -> (hBOOL(summable_real(A)) -> (exists N_3 all N_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_3),N_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_f352196356l_real(suminf_real,hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,A),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),N_2))))),R_2))))))) # label(fact_4852_suminf__exist__split) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1564 (all M_27 all A_224 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_224),one_on1684967323de_int)),M_27) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,M_27),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_224),M_27))) # label(fact_404_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1565 (all C_74 all A_155 all B_131 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_155),B_131)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_74)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_74),A_155)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_74),B_131)))))) # label(fact_1210_mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1566 (all N_33 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_int_rat(number_number_of_rat,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_33)) = one_one_rat) # label(fact_1172_power__m1__even) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1567 (all Y_1 all X_1 all U_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,X_1)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,U_1),hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,Y_1)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,U_1),hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),U_1))))) # label(fact_3378_real__sqrt__sum__squares__less) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1568 (all N_54 all A_204 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_204)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_204),N_54))))) # label(fact_660_zero__less__power) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1569 (all B_112 all C_67 all A_134 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_134)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_112),C_67)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_112),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_134),C_67)))))) # label(fact_1333_pos__add__strict) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1570 (all B_1_1 all B_2_1 is_int(big_co1548731110nt_int(B_1_1,B_2_1))) # label(gsy_c_Big__Operators_Ocomm__monoid__mult__class_Osetprod_000tc__Int__Oint_000tc_) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1571 (all P_1 P_1 = collect_real(P_1)) # label(fact_210_Collect__def) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1572 (all A (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),A)),zero_zero_real)))) # label(fact_2197_double__add__less__zero__iff__single__add__less__zero) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1573 (all N_1 (is_int(N_1) -> archim856651990g_real(hAPP_int_real(real_int,N_1)) = N_1)) # label(fact_4396_ceiling__real__of__int) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1574 (all I all M all N_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N_1))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),M)))) # label(fact_4983_dvd__gcd__D1__int) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1575 (all N_66 all A_217 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_217)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_217),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_217),N_66)))))) # label(fact_526_power__gt1__lemma) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1576 (all Y_2 all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(ln,X_2)),hAPP_real_real(ln,Y_2))))))) # label(fact_3426_ln__less__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1577 (all X_2 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zOdd)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) -> hAPP_int_int(number_number_of_int,min) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_nat(nat,X_2))))) # label(fact_2767_EvenOdd_Oneg__one__odd__power) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1578 (all C (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,C)),one_one_real)) -> hBOOL(tendsto_nat_real(hAPP_r474017924t_real(power_power_real,C),zero_zero_real,sequentially)))) # label(fact_5037_LIMSEQ__rabs__realpow__zero2) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1579 (all A_235 all B_164 all C_95 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_235),B_164)),C_95) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_235),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_164),C_95))) # label(fact_223_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1580 (all A_33 all B_29 all C_12 ((zero_z126310315umeral = C_12 -> hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_12),A_33)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_12),B_29)) = zero_z126310315umeral) & (C_12 != zero_z126310315umeral -> hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_33),B_29) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_12),A_33)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_12),B_29))))) # label(fact_2567_div__mult__mult1__if) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1581 (all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B)),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)))) # label(fact_2215_le__iff__diff__le__0) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1582 (all X_1 all Y_1 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(legacy_zgcd(X_1),Y_1)))) # label(fact_4498_zgcd__geq__zero) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1583 (all A_3 (is_int(A_3) -> A_3 = hAPP_int_int(div_mod_int(A_3),zero_zero_int) & zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),zero_zero_int))) # label(fact_2561_DIVISION__BY__ZERO) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1584 (all A_201 all N_50 all N_49 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_50),N_49)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_201)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_201),one_one_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_201),N_49)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_201),N_50))))))) # label(fact_681_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1585 (all Z_2 all W_1 (W_1 != Z_2 & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Z_2),W_1)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Z_2),W_1)))) # label(fact_4656_less__rat__def) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1586 (all X_1 all Y_1 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_1)),X_1))) # label(fact_4959_gcd__dvd1__int) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1587 (all A_226 all B_161 all V_18 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_226),B_161)),hAPP_int_complex(number528085621omplex,V_18)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_226),hAPP_int_complex(number528085621omplex,V_18))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_161),hAPP_int_complex(number528085621omplex,V_18)))) # label(fact_382_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1588 (all N all V hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))),hAPP_nat_nat(suc,N)),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,N),hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(suc,N)),hAPP_int_nat(number_number_of_nat,V))) # label(fact_5183_max__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1589 (all Y_2 all X_2 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),X_2)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),Y_2)) -> (zero_z891286103de_int = Y_2 & zero_z891286103de_int = X_2 <-> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X_2),Y_2) = zero_z891286103de_int)))) # label(fact_2170_add__nonneg__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1590 (all A_58 all C_21 all B_50 hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_58),B_50)),C_21) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(div_mod_nat(A_58),C_21)),B_50)),C_21)) # label(fact_2367_zmod__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1591 (all A_3 all B_2 (is_int(B_2) -> (B_2 != zero_zero_int -> (zero_zero_int != hAPP_int_int(div_mod_int(A_3),B_2) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(uminus_uminus_int,A_3)),B_2) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2))),one_one_int)) & (zero_zero_int = hAPP_int_int(div_mod_int(A_3),B_2) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(uminus_uminus_int,A_3)),B_2) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)))))) # label(fact_3551_zdiv__zminus1__eq__if) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1592 (all A_3 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,one_one_nat),A_3))) # label(fact_2282_gcd__lcm__complete__lattice__nat_Obot__least) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1593 (all X_1 all Y_1 hAPP_complex_complex(cnj,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_1),Y_1)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(cnj,X_1)),hAPP_complex_complex(cnj,Y_1))) # label(fact_4067_complex__cnj__mult) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1594 (all N_1 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),N_1)) -> hAPP_int_int(legacy_zgcd(N_1),P_3) = one_one_int))) # label(fact_4520_zprime__imp__zrelprime) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1595 (all P all Q all R hAPP_f428220345t_bool(P,hAPP_i345030060t_bool(Q,R)) = hAPP_i345030060t_bool(hAPP_f791698359t_bool(hAPP_f847817363t_bool(cOMBB_562416518ol_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__HOL__026) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1596 (all Q_2 ((zero_zero_rat != Q_2 -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),Q_2)) -> hAPP_rat_rat(sgn_sgn_rat,Q_2) = one_one_rat) & (-hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),Q_2)) -> hAPP_rat_rat(uminus_uminus_rat,one_one_rat) = hAPP_rat_rat(sgn_sgn_rat,Q_2))) & (Q_2 = zero_zero_rat -> zero_zero_rat = hAPP_rat_rat(sgn_sgn_rat,Q_2)))) # label(fact_4649_sgn__rat__def) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1597 (all Ma all N (is_int(Ma) & is_int(N) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Ma),N))) <-> N != zero_zero_int | Ma != zero_zero_int))) # label(fact_4997_gcd__pos__int) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1598 (all K_1 all Ma all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),hAPP_int_int(legacy_zgcd(Ma),N))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),N)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),Ma)))) # label(fact_4488_zgcd__greatest__iff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1599 (all C_57 all A_123 all B_102 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_57),A_123)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_57),B_102))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_57)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_123),B_102))))) # label(fact_1464_mult__left__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1600 (all A_163 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,one_one_rat),A_163))) # label(fact_1152_one__dvd) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1601 (all A_44 all B_37 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_44),B_37)) -> zero_z126310315umeral = hAPP_C498520661umeral(div_mo1740067990umeral(B_37),A_44))) # label(fact_2427_dvd__imp__mod__0) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1602 (all C_80 all A_164 all B_139 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_164),B_139)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_164),C_80)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_164),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_139),C_80)))))) # label(fact_1141_dvd__add) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1603 (all Z_10 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Z_10),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Z_10),Z_10)) # label(fact_497_semiring__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1604 (all P all Q all R hAPP_bool_bool(hAPP_P921862901l_bool(P,R),hAPP_P603027463t_bool(Q,R)) = hAPP_P603027463t_bool(hAPP_f237669397t_bool(hAPP_f461489489t_bool(cOMBS_400904860l_bool,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__HOL__Obo) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1605 (all N_1 hAPP_complex_real(im,hAPP_nat_complex(semiri2020571505omplex,N_1)) = zero_zero_real) # label(fact_4101_complex__Im__of__nat) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1606 (all A (zero_zero_rat = A <-> zero_zero_rat = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),A))) # label(fact_643_double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1607 (all B_150 all A_178 (is_int(B_150) & is_int(A_178) -> (A_178 != zero_zero_int -> (B_150 != zero_zero_int -> zero_zero_int != hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_178),B_150))))) # label(fact_1025_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1608 (all Y_1 all X_1 (hBOOL(hAPP_int_bool(nat_is_nat,X_1)) -> (hBOOL(hAPP_int_bool(nat_is_nat,Y_1)) -> hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),Y_1)))))) # label(fact_5136_Divides_Otransfer__int__nat__function__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1609 (all X_1 hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),pi)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_1))) # label(fact_3668_sin__periodic__pi) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1610 (all Ma all N all P_1 ((all M_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,M_1),zero_zero_nat))) -> ((all M_1 all N_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N_2),hAPP_nat_nat(div_mod_nat(M_1),N_2))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,M_1),N_2))))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Ma),N))))) # label(fact_4550_gcd__nat__induct) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1611 (all D all B all C all A (is_int(C) & is_int(A) -> (A != zero_zero_int -> (zero_zero_int != C -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),B)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,C),D)) -> (hAPP_int_int(hAPP_int_fun_int_int(div_div_int,D),C) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B),A) <-> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),C) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),D)))))))) # label(fact_2661_dvd__div__div__eq__mult) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1612 (all V_17 all B_160 all C_90 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_17)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_160),C_90)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_17)),B_160)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_17)),C_90))) # label(fact_387_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1613 (all X_2 all P_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_5),X_2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_2),zero_zero_int),P_5)))) # label(fact_1399_zcong__eq__zdvd__prop) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1614 (all V_11 all W_9 hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_11),W_9)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_11)),hAPP_int_complex(number528085621omplex,W_9))) # label(fact_443_number__of__add) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1615 (all R_3 all X_1 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_complex_real(im,X_1)) = hAPP_complex_real(im,hAPP_complex_complex(scaleR1652505878omplex(R_3),X_1))) # label(fact_5107_complex__Im__scaleR) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1616 (all A_236 all B_165 all C_96 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_236),B_165)),C_96) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_236),C_96)),B_165)) # label(fact_214_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1617 (all A_22 all C_7 all B_19 (is_int(B_19) -> (zero_zero_int != B_19 -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_22),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_19),C_7))),B_19) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_7),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_22),B_19))))) # label(fact_2643_div__mult__self2) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1618 (all A_127 all C_61 all B_106 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_127),C_61)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_106),C_61))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_61)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_127),B_106))))) # label(fact_1440_mult__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1619 (all A_3 all B_2 normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_3),B_2)) = quotient_of(hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2))) # label(fact_4710_quotient__of__Fract) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1620 (all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),N_1) = zero_zero_nat) # label(fact_5122_min__0L) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1621 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(fact,M)),hAPP_nat_nat(fact,N_1))))) # label(fact_4558_fact__mono) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1622 (all A_49 all B_41 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_49),B_41)),B_41) = zero_zero_int) # label(fact_2407_mod__mult__self2__is__0) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1623 (all X_1 all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),hAPP_int_int(number_number_of_int,min)),P_3)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),one_one_int),P_3))))) # label(fact_1007_zcong__neg__1__impl__ne__1) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1624 (all Z_7 all W_2 all Y_10 all X_14 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,Y_10),X_14)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,Z_7),W_2)) -> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,X_14),Y_10)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,W_2),Z_7)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_14),W_2)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Y_10),Z_7))))) # label(fact_2606_div__mult__div__if__dvd) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1625 (all X_2 all F ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(F,N_2)))) -> ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_2))),X_2))) -> hBOOL(summable_real(F))))) # label(fact_4945_pos__summable) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1626 (all X_3 all A (hBOOL(tendsto_nat_complex(X_3,A,sequentially)) -> hBOOL(tendsto_nat_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,re),X_3),hAPP_complex_real(re,A),sequentially)))) # label(fact_5055_Re_OLIMSEQ) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1627 (all X_47 all P_8 all Q_12 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_47),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_8),Q_12)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_47),P_8)),Q_12)) # label(fact_72_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1628 (all A_189 all N_40 all N_39 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_40),N_39)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),A_189)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_189),N_40)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_189),N_39)))))) # label(fact_868_power__increasing) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1629 (all A_239 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_239),A_239) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_239),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_41_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1630 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),B_2)) -> zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)))) # label(fact_2628_div__pos__pos__trivial) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1631 (all Ma all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) | Ma = N)) # label(fact_1770_le__eq__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1632 (all N all X_2 all Diff all F ((all M_1 all X hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),X),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),X)))) & F = hAPP_n546711566l_real(Diff,zero_zero_nat) -> (exists T (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,X_2))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),N))) = hAPP_real_real(F,X_2) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_2))))))) # label(fact_4940_Maclaurin__all__le__objl) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1633 (all Y_2 all X_2 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven)) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_2),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,Y_2),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Y_2)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_3239_even__sum__div__2) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1634 (all J_2 all A_3 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),zero_zero_int),P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),A_3)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),J_2))),P_3)))))))) # label(fact_1893_MultInvPair__distinct) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1635 (all R_3 all Q_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (A_3 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,R_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),Q_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),A_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),Q_2)))))) # label(fact_904_self__quotient__aux1) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1636 (all X_2 all Y_2 (zero_zero_rat = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_2),X_2)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_2),Y_2)) <-> Y_2 = zero_zero_rat & zero_zero_rat = X_2)) # label(fact_692_sum__squares__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1637 (all A all B (is_int(B) & is_int(A) -> (hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B) = zero_zero_int <-> B = A))) # label(fact_2090_right__minus__eq) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1638 (all K_1 hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_eq_nat),K_1))))) # label(fact_4731_finite__Collect__le__nat) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1639 (all N_11 all A_6 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_6)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_6),one_one_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_6),hAPP_nat_nat(suc,N_11))),one_one_rat))))) # label(fact_3162_power__Suc__less__one) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1640 (all N_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N_1))) # label(fact_1632_le0) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1641 (all X_30 all N_34 (X_30 = one_one_complex | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_34)) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,X_30),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_30),N_34))))) # label(fact_994_dvd__power) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1642 (all W_7 hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,W_7)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat)),hAPP_int_rat(number_number_of_rat,W_7))) # label(fact_478_double__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1643 (all Z_1 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,Z_1)))) # label(fact_2989_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1644 (all I_1 all P_1 all K_1 (hBOOL(hAPP_int_bool(P_1,K_1)) -> ((all I_2 (is_int(I_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),I_2)) -> (hBOOL(hAPP_int_bool(P_1,I_2)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I_2),one_one_int))))))) -> ((all I_2 (is_int(I_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),K_1)) -> (hBOOL(hAPP_int_bool(P_1,I_2)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,I_2),one_one_int))))))) -> hBOOL(hAPP_int_bool(P_1,I_1)))))) # label(fact_4004_int__induct) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1645 (all X_9 all Q_7 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_9),Q_7)),X_9) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_9),hAPP_nat_nat(suc,Q_7))) # label(fact_3006_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1646 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K1)),hAPP_int_int(bit1,K2))))) # label(fact_129_less__int__code_I14_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1647 (all C_46 all A_103 all B_86 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_46),A_103)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_46),B_86))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_103),B_86)))) # label(fact_1995_add__le__imp__le__left) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1648 (all B_113 all A_136 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_136)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_113)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_136),B_113)))))) # label(fact_1317_mult__pos__pos) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1649 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B)),zero_zero_rat)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),zero_zero_int))))) # label(fact_4723_Fract__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1650 (all A_19 all B_16 (B_16 != zero_z126310315umeral -> hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_19),B_16)),one_on1645066479umeral) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_16),A_19)),B_16))) # label(fact_2657_div__add__self1) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1651 (all X_2 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),Y_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),X_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(tan,Y_2)),hAPP_real_real(tan,X_2))))))))) # label(fact_3615_tan__monotone_H) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1652 (all A_195 all N_47 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_195),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_195),N_47)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_195),N_47)),A_195)) # label(fact_806_power__commutes) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1653 (all A_108 all B_87 all C_47 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_108),B_87)),C_47) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_108),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_87),C_47))) # label(fact_1949_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1654 (all X_1 all N_1 (-hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1)) -> hAPP_real_real(hAPP_n546711566l_real(root,N_1),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1)) = X_1)) # label(fact_3898_odd__real__root__power__cancel) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1655 (all A_3 all B_2 hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_3),B_2)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_nat_int(semiri1621563631at_int,A_3)),hAPP_nat_int(semiri1621563631at_int,B_2))) # label(fact_3077_zdiv__int) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1656 (all A_175 all E_3 all B_149 all C_87 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_175),B_149)),E_3)),C_87) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_175),E_3)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_149),E_3)),C_87))) # label(fact_1067_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1657 (all K_2 ((zero_z891286103de_int = K_2 -> quickcheck_int_of(K_2) = zero_zero_int) & (K_2 != zero_z891286103de_int -> (zero_z891286103de_int != hAPP_Q1762011733de_int(div_mo231679042de_int(K_2),hAPP_i1732201573de_int(number1226105091de_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) -> quickcheck_int_of(K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),quickcheck_int_of(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,K_2),hAPP_i1732201573de_int(number1226105091de_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),one_one_int)) & (hAPP_Q1762011733de_int(div_mo231679042de_int(K_2),hAPP_i1732201573de_int(number1226105091de_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_z891286103de_int -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),quickcheck_int_of(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,K_2),hAPP_i1732201573de_int(number1226105091de_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = quickcheck_int_of(K_2))))) # label(fact_4249_Quickcheck__Narrowing_Oint__of__code) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1658 (all B_72 all C_35 all A_83 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_83)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_72),C_35)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_72),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_83),C_35)))))) # label(fact_2160_add__increasing) # label(axiom) # label(non_clause). [assumption]. 5.43/5.46 1659 (all A_3 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natfloor(X_1)),A_3) = natfloor(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_nat_real(real_nat,A_3))))) # label(fact_3701_natfloor__add) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1660 (all A all E_2 all C all B all D (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),E_2)),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),E_2)),D))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B),A)),E_2)),D))))) # label(fact_1570_le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1661 (all Z_1 zero_zero_real = hAPP_complex_real(im,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Z_1),hAPP_complex_complex(cnj,Z_1)))) # label(fact_4103_complex__In__mult__cnj__zero) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1662 (all A_25 all B_22 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_25),B_22)) -> B_22 = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,B_22),A_25)),A_25))) # label(fact_2597_dvd__div__mult__self) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1663 (all N_1 all X_1 (hBOOL(hAPP_int_bool(nat_is_nat,X_1)) -> hBOOL(hAPP_int_bool(nat_is_nat,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),N_1))))) # label(fact_5134_Nat__Transfer_Otransfer__int__nat__function__closures_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1664 (all K_2 hAPP_int_int(succ,hAPP_int_int(bit0,K_2)) = hAPP_int_int(bit1,K_2)) # label(fact_3296_succ__Bit0) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1665 (all X_1 all Y_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),hAPP_real_real(hAPP_n546711566l_real(root,N_1),Y_1)) = hAPP_real_real(hAPP_n546711566l_real(root,N_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_1),Y_1)))) # label(fact_3911_real__root__divide) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1666 (all R_3 all A_3 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,R_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_real_real(cos,A_3))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_real_real(sin,A_3))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3726_sin__cos__squared__add2__mult) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1667 (all B_1_1 all B_2_1 is_int(hAPP_P1175774780nt_int(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oint) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1668 (all A_210 zero_z126310315umeral = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,zero_z126310315umeral),A_210)) # label(fact_611_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1669 (all V_14 all W_12 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_14)),hAPP_int_rat(number_number_of_rat,W_12)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_14),W_12))) # label(fact_432_number__of__mult) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1670 (all C all A all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A),B)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C),A)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C),B))))) # label(fact_2026_add__le__cancel__left) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1671 (all L_2 hAPP_int_int(bit1,L_2) != pls) # label(fact_349_rel__simps_I39_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1672 (all C_70 all A_145 all B_122 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_145),B_122)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_70)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_70),A_145)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_70),B_122)))))) # label(fact_1261_comm__mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1673 (all Y_1 (zero_zero_real != Y_1 -> hAPP_real_real(cos,arg(hAPP_real_complex(hAPP_r265291036omplex(complex,zero_zero_real),Y_1))) = zero_zero_real)) # label(fact_3972_cos__arg__i__mult__zero) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1674 (all P all Q all R hAPP_nat_int(hAPP_int_fun_nat_int(hAPP_f1673103573at_int(cOMBC_nat_int_int,P),Q),R) = hAPP_int_int(hAPP_nat_fun_int_int(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Int__Oint_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1675 (all K_1 all L (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(pred,K_1)),L)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),L)))) # label(fact_4041_le__iff__pred__less) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1676 (all A_26 all C_9 all B_23 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C_9),B_23)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_26),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B_23),C_9)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_26),B_23)),C_9))) # label(fact_2596_div__mult__swap) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1677 (all N (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N)),zero_zero_real)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,N),zero_zero_int)))) # label(fact_4437_real__of__int__lt__zero__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1678 (all V_11 all W_9 hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_11),W_9)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_11)),hAPP_int_rat(number_number_of_rat,W_9))) # label(fact_444_number__of__add) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1679 (all Diff all F all N all H (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) & hAPP_n546711566l_real(Diff,zero_zero_nat) = F & (all M_1 all T (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,H),T)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),zero_zero_real)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N)) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),T),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T))))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H),zero_zero_real)) -> (exists T (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H),T)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,T),zero_zero_real)) & hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,H))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),N))) = hAPP_real_real(F,H))))) # label(fact_4937_Maclaurin__minus__objl) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1680 (all X_2 all A_1 (hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))))) # label(fact_4613_Nat__Transfer_Otransfer__nat__int__set__function__closures_I6_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1681 (all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,pls)),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_int_real(number267125858f_real,Y_2))))) # label(fact_294_le__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1682 (all A_225 all M_28 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_225),one_on1684967323de_int)),M_28) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_225),M_28)),M_28)) # label(fact_397_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1683 (all X_2 hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),cos)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,min))),hAPP_real_real(cos,X_2))),hAPP_real_real(sin,X_2))))) # label(fact_4796_DERIV__cos__realpow2a) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1684 (all A_177 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_177),zero_zero_real) = zero_zero_real) # label(fact_1038_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1685 (all D all Sa exists Z all X (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Z),X)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X),Sa))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X),Sa)))))) # label(fact_2248_pinf_I9_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1686 (all A_108 all B_87 all C_47 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_108),B_87)),C_47) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_108),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_87),C_47))) # label(fact_1951_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1687 (all Y_1 all X_1 (hAPP_real_real(exp_real,Y_1) = X_1 -> hAPP_real_real(ln,X_1) = Y_1)) # label(fact_3819_ln__unique) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1688 (all N_1 hAPP_nat_nat(suc,zero_zero_nat) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(suc,zero_zero_nat)),N_1)) # label(fact_3085_power__Suc__0) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1689 (all X_3 ((all R_1 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),R_1)) -> (exists K all N_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N_2)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,hAPP_nat_rat(X_3,N_2))),R_1)))))) -> hBOOL(vanishes(X_3)))) # label(fact_5151_vanishesI) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1690 (all X_6 all Y_5 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_6),Y_5)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_6),Y_5)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_6),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_5),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))))) # label(fact_3194_realpow__two__diff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1691 (all Lx_3 all Ly_1 all Rx_3 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_3),Rx_3)),Ly_1) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_3),Ly_1)),Rx_3)) # label(fact_162_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1692 (all B_118 all A_141 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_141),zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_118)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_141),B_118)),zero_zero_nat))))) # label(fact_1282_mult__neg__pos) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1693 (all A_3 ((-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),zero_zero_real)) -> hAPP_real_real(abs_abs_real,A_3) = A_3) & (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),zero_zero_real)) -> hAPP_real_real(abs_abs_real,A_3) = hAPP_real_real(uminus_uminus_real,A_3)))) # label(fact_3494_abs__real__def) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1694 (all Y_22 all X_28 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_28)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_22)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_22),one_one_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_22),X_28)),X_28)))))) # label(fact_1467_mult__left__le__one__le) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1695 (all Lx all Rx all Ry hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx),Ry)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx),Ry))) # label(fact_182_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1696 (all B_73 all A_84 all C_36 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_36)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_73),A_84)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_73),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_84),C_36)))))) # label(fact_2154_add__increasing2) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1697 (all X_2 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma)) -> (one_one_int = hAPP_int_int(legacy_zgcd(X_2),Ma) -> hBOOL(hAPP_int_bool(is_RRset(hAPP_i1948725293t_bool(noXRRset(Ma),X_2)),Ma))))) # label(fact_4587_noX__is__RRset) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1698 (all N_60 all X_41 all Y_33 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,X_41),Y_33)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_41),N_60)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_33),N_60))))) # label(fact_585_dvd__power__same) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1699 (all Z_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),Z_2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_2)))) # label(fact_753_int__one__le__iff__zero__less) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1700 (all X_1 all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,X_1)),A_3)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),natfloor(A_3))))) # label(fact_3674_le__natfloor) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1701 (all K_2 (is_int(K_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2)) -> (exists N_2 hAPP_nat_int(semiri1621563631at_int,N_2) = K_2)))) # label(fact_3259_zero__le__imp__eq__int) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1702 (all A_106 A_106 = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_106),zero_z126310315umeral)) # label(fact_1970_add__0__right) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1703 (all K_2 frct(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),K_2)) = zero_zero_rat) # label(fact_4641_Frct__code__post_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1704 (all P all Q all R hAPP_nat_real(hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,P),Q),R) = hAPP_real_real(hAPP_n546711566l_real(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__RealDef__Orea) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1705 (all X_1 hAPP_real_real(uminus_uminus_real,hAPP_real_real(sqrt,X_1)) = hAPP_real_real(sqrt,hAPP_real_real(uminus_uminus_real,X_1))) # label(fact_3479_real__sqrt__minus) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1706 (all N_1 all M hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1)))) # label(fact_1783_le__add2) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1707 (all C_38 all A_95 all B_78 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_95),B_78)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_38),A_95)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_38),B_78))))) # label(fact_2059_add__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1708 (all C_70 all A_145 all B_122 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_145),B_122)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_70)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_70),A_145)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_70),B_122)))))) # label(fact_1262_comm__mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1709 (all P_3 ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_P1175774780nt_int(product_snd_int_int,P_3))) -> (hAPP_P1175774780nt_int(product_snd_int_int,P_3) = zero_zero_int -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int) = normalize(P_3)) & (hAPP_P1175774780nt_int(product_snd_int_int,P_3) != zero_zero_int -> normalize(P_3) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_P1175774780nt_int(product_snd_int_int,P_3))))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_P1175774780nt_int(product_snd_int_int,P_3)),hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_P1175774780nt_int(product_snd_int_int,P_3))))))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_P1175774780nt_int(product_snd_int_int,P_3))) -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_P1175774780nt_int(product_snd_int_int,P_3)))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_P1175774780nt_int(product_snd_int_int,P_3)),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_P1175774780nt_int(product_snd_int_int,P_3)))) = normalize(P_3)))) # label(fact_4958_normalize__def) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1710 (all X_2 hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,fact_fact_nat),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_2)),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(cos,X_2)))) # label(fact_4869_cos__paired) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1711 (all C_60 all A_126 all B_105 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_60),A_126)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_60),B_105))) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_60)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_126),B_105))))) # label(fact_1448_mult__left__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1712 (all M hAPP_int_int(abs_abs_int,hAPP_nat_int(semiri1621563631at_int,M)) = hAPP_nat_int(semiri1621563631at_int,M)) # label(fact_3416_abs__int__eq) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1713 (all N_9 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,N_9)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_9)))) # label(fact_3216_of__nat__less__two__power) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1714 (all X_2 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sqrt,X_2)),hAPP_real_real(sqrt,Y_2))))) # label(fact_3312_real__sqrt__le__iff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1715 (all M_9 all N_23 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(semiri151668891at_rat,M_9)),hAPP_nat_rat(semiri151668891at_rat,N_23))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_23)))) # label(fact_2944_of__nat__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1716 (all B_2 all Y_1 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_1)) -> hAPP_real_real(log(B_2),hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_1),hAPP_real_real(log(B_2),X_1))))) # label(fact_4472_log__powr) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1717 (all C_88 all A_180 all B_152 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_180),B_152)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_152),C_88)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_180),C_88))))) # label(fact_991_dvd__trans) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1718 (all X_1 hAPP_int_int(abs_abs_int,X_1) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),X_1)) # label(fact_4988_gcd__idem__int) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1719 (all X_2 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_2),N)),zOdd)) <-> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zOdd))))) # label(fact_2688_power__preserves__odd) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1720 (all A_21 all C_6 all B_18 (zero_z126310315umeral != B_18 -> hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_21),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_6),B_18))),B_18) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_6),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_21),B_18)))) # label(fact_2649_div__mult__self1) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1721 (all C_88 all A_180 all B_152 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_180),B_152)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,B_152),C_88)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_180),C_88))))) # label(fact_986_dvd__trans) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1722 (all K_2 all M all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N_1))) # label(fact_4333_gcd__mult__distrib__nat) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1723 (all B_14 all A_17 A_17 = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_14),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_17),B_14))),hAPP_C498520661umeral(div_mo1740067990umeral(A_17),B_14))) # label(fact_2673_mod__div__equality2) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1724 (all X_44 all Y_36 all Z_15 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_44),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,Y_36),Z_15)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_44),Y_36)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_44),Z_15))) # label(fact_325_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1725 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,X_1)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X_1)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))))) # label(fact_3855_exp__bound) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1726 (all F all L all C (hBOOL(tendsto_real_real(F,L,at_real(C))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),L)) -> (exists R_1 ((all X (X != C & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),X))),R_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(F,X))))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_1))))))) # label(fact_5086_LIM__fun__gt__zero) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1727 (all A_209 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_209),zero_zero_real) = zero_zero_real) # label(fact_617_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1728 (all N_54 all A_204 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_204)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_204),N_54))))) # label(fact_661_zero__less__power) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1729 (all A (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),A))))) # label(fact_2201_zero__less__double__add__iff__zero__less__single__add) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1730 (all Q_2 all R_3 hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,Q_2),R_3) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Q_2),hAPP_rat_rat(uminus_uminus_rat,R_3))) # label(fact_4655_diff__rat__def) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1731 (all A all B all W_1 (hAPP_real_complex(hAPP_r265291036omplex(complex,A),B) = hAPP_int_complex(number528085621omplex,W_1) <-> hAPP_int_real(number267125858f_real,W_1) = A & zero_zero_real = B)) # label(fact_3958_Complex__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1732 (all X_2 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,ratreal(X_2)),ratreal(Y_2))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_2),Y_2)))) # label(fact_4699_real__less__eq__code) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1733 (all A_107 (is_int(A_107) -> A_107 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_107),zero_zero_int))) # label(fact_1964_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1734 (all A_92 hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_92),A_92) = zero_zero_complex) # label(fact_2084_diff__self) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1735 (all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_int_real(real_int,N))))) # label(fact_4440_real__of__int__ge__zero__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1736 (all A_176 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,zero_zero_real),A_176) = zero_zero_real) # label(fact_1045_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1737 (all A hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(d22set,A)))) # label(fact_3358_d22set__fin) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1738 (all P all Q all R hAPP_int_rat(hAPP_f1900950721nt_rat(hAPP_f1270196437nt_rat(cOMBB_int_rat_int,P),Q),R) = hAPP_int_rat(P,hAPP_int_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__Rat__Orat_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1739 (all X_1 all Y_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),hAPP_real_real(hAPP_n546711566l_real(root,N_1),Y_1)))))) # label(fact_3904_real__root__less__mono) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1740 (all N_1 all P_3 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N_1))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),N_1)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),M)))))) # label(fact_909_zprime__zdvd__zmult) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1741 (all P all Q all R hAPP_r1250527377l_real(hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,P),Q),R) = hAPP_r1250527377l_real(P,hAPP_real_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__R_011) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1742 (all X_1 hAPP_real_real(cos,hAPP_real_real(uminus_uminus_real,X_1)) = hAPP_real_real(cos,X_1)) # label(fact_3560_cos__minus) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1743 (all A_201 all N_50 all N_49 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_50),N_49)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_201)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_201),one_on1645066479umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_201),N_49)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_201),N_50))))))) # label(fact_684_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1744 (all X_3 (hBOOL(summable_complex(X_3)) -> hAPP_complex_real(re,hAPP_f213450978omplex(suminf_complex,X_3)) = hAPP_f352196356l_real(suminf_real,hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,re),X_3)))) # label(fact_4847_Re_Osuminf) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1745 (all M all N_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N_1)),M))) # label(fact_2717_div__le__dividend) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1746 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) -> hAPP_nat_nat(div_mod_nat(M),N_1) = M)) # label(fact_2507_mod__less) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1747 (all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_1)) -> (exists X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) & hAPP_real_real(tan,X) = Y_1 & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))))) # label(fact_3625_tan__total__pos) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1748 (all C_75 all A_156 all B_132 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_156),B_132)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_75)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_75),A_156)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_75),B_132)))))) # label(fact_1203_comm__mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1749 (all Real1 all Real2 hAPP_complex_nat(size_size_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,Real1),Real2)) = zero_zero_nat) # label(fact_3970_complex_Osize_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1750 (all B_1_1 is_int(archim1246769320r_real(B_1_1))) # label(gsy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_000tc__RealDef__Oreal) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1751 (all P all Q all R hAPP_r474017924t_real(hAPP_f1501741374t_real(hAPP_f1135648319t_real(cOMBC_1329014627t_real,P),Q),R) = hAPP_f1676298106t_real(hAPP_r50993370t_real(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Nat__Onat_Mtc__Nat__O) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1752 (all A_210 zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,zero_zero_int),A_210)) # label(fact_608_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1753 (all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),B_2)) -> (B_2 != one_one_nat -> (exists X exists Y hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),X) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),Y)),one_one_nat))))) # label(fact_4036_coprime__bezout__strong) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1754 (all Z_1 zero_zero_real = hAPP_complex_real(im,hAPP_int_complex(ring_11397209091omplex,Z_1))) # label(fact_4583_complex__Im__of__int) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1755 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K1)),hAPP_int_int(bit0,K2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)))) # label(fact_113_less__int__code_I13_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1756 (all X_1 all Y_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3350_real__sqrt__sum__squares__ge__zero) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1757 (all B_1_1 is_bool(summable_complex(B_1_1))) # label(gsy_c_Series_Osummable_000tc__Complex__Ocomplex) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1758 (all N_25 hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),hAPP_n522471361de_int(semiri1424489471de_int,N_25)))) # label(fact_2907_of__nat__0__le__iff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1759 (all X_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),one_one_nat) = one_one_nat) # label(fact_4336_gcd__lcm__complete__lattice__nat_Oinf__bot__right) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1760 (all A_33 all B_29 all C_12 ((C_12 = zero_zero_nat -> zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_12),A_33)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_12),B_29))) & (C_12 != zero_zero_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_33),B_29) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_12),A_33)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_12),B_29))))) # label(fact_2566_div__mult__mult1__if) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1761 (all A all B all C (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),A)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B)))))) # label(fact_1321_mult__less__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1762 (all M_11 -hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(semiri151668891at_rat,M_11)),zero_zero_rat))) # label(fact_2918_of__nat__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1763 (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_2)) <-> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sqrt,X_2)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = X_2)) # label(fact_3342_real__sqrt__pow2__iff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1764 (all P_1 ((all X exists D_2 ((all A_2 all B_4 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),B_4)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_4),A_2)),D_2)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),X)) -> hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),B_4))))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2)))) & (all A_2 all B_4 all C_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_4),C_3)) & hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,B_4),C_3))) & hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),B_4))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),B_4)) -> hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),C_3))))) -> (all A_2 all B_4 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),B_4)) -> hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),B_4))))))) # label(fact_4290_lemma__BOLZANO2) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1765 (all C_68 all A_143 all B_120 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_143),B_120)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_68)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_143),C_68)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_120),C_68)))))) # label(fact_1273_mult__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1766 (all X_8 all Q_6 hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_8),hAPP_nat_nat(suc,Q_6)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,X_8),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_8),Q_6))) # label(fact_3014_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1767 (all A_179 all B_151 (zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_179),B_151) -> zero_zero_nat = B_151 | A_179 = zero_zero_nat)) # label(fact_1019_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1768 (all K_2 all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),N_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),M)))))) # label(fact_1668_dvd__diffD) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1769 (all A_170 all B_145 all K_7 (A_170 = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_145),K_7) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,B_145),A_170)))) # label(fact_1101_dvdI) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1770 (all X_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(cos,X_1))),one_one_real))) # label(fact_3567_abs__cos__le__one) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1771 (all Y_1 all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Y_1),X_1)) -> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_1) = hAPP_int_int(abs_abs_int,Y_1))) # label(fact_4995_gcd__proj2__if__dvd__int) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1772 (all A_18 all B_15 A_18 = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_18),B_15)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_18),B_15)),B_15))) # label(fact_2670_semiring__div__class_Omod__div__equality_H) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1773 (all X_2 all Y_2 (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_2),Y_2)) -> (X_2 != Y_2 -> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,X_2),Y_2))))) # label(fact_1016_order__le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1774 (all D (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,zero_zero_nat),D)) <-> one_one_nat = D)) # label(fact_4023_coprime__0_H) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1775 (all M all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),M))))) # label(fact_1763_trans__less__add1) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1776 (all A hBOOL(hAPP_complex_bool(isCont156215680omplex(cnj),A))) # label(fact_5164_cnj_OisCont) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1777 (all N_1 all X_1 (X_1 = hAPP_nat_real(real_nat,natfloor(X_1)) -> natfloor(hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,natfloor(X_1)),N_1))) # label(fact_3677_natfloor__power) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1778 (all X_1 ratreal(hAPP_rat_rat(inverse_inverse_rat,X_1)) = hAPP_real_real(inverse_inverse_real,ratreal(X_1))) # label(fact_4705_real__inverse__code) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1779 (all R_3 all A_3 all B_2 hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),A_3)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),B_2)) = hAPP_complex_complex(scaleR1652505878omplex(R_3),hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),B_2))) # label(fact_5109_complex__scaleR) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1780 (all X_2 all K_1 (X_2 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),K_1)),one_one_int) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zOdd)))) # label(fact_2532_zOddI) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1781 (all M all N_1 hAPP_int_int(legacy_zgcd(N_1),hAPP_int_int(div_mod_int(M),N_1)) = hAPP_int_int(legacy_zgcd(M),N_1)) # label(fact_4484_zgcd__eq) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1782 (all C_63 all D_10 all A_129 all B_108 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_129),B_108)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_63),D_10)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_129)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_63)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_129),C_63)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_108),D_10)))))))) # label(fact_1426_mult__le__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1783 (all I all J_2 hAPP_int_int(legacy_zgcd(hAPP_int_int(uminus_uminus_int,I)),J_2) = hAPP_int_int(legacy_zgcd(I),J_2)) # label(fact_4494_zgcd__zminus) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1784 (all A (hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially)) -> ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(A,N_2)))) -> ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A,hAPP_nat_nat(suc,N_2))),hAPP_nat_real(A,N_2)))) -> hBOOL(tendsto_nat_real(hAPP_f292751352t_real(hAPP_f1607377819t_real(cOMBB_1533963633al_nat,hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A))),hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,ord_at4362885an_nat(zero_zero_nat)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),sequentially)))))) # label(fact_5052_summable__Leibniz_H_I5_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1785 (all B_73 all A_84 all C_36 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_36)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_73),A_84)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_73),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_84),C_36)))))) # label(fact_2155_add__increasing2) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1786 (all W_1 all Z_2 (is_int(W_1) & is_int(Z_2) -> (W_1 = Z_2 | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_2),one_one_int)))))) # label(fact_276_zless__add1__eq) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1787 (all A_3 all B_2 all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),B_2)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),A_3))))) # label(fact_4169_prime__divprod) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1788 (all K_2 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,min),K_2) = hAPP_int_int(uminus_uminus_int,K_2)) # label(fact_3534_mult__Min) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1789 (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) -> hAPP_real_real(ln,X_2) = hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,one_one_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),one_one_nat)))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_2),one_one_real))),suc)))))) # label(fact_4825_ln__series) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1790 (all C_37 all A_94 all B_77 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_94),B_77)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_94),C_37)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_77),C_37))))) # label(fact_2063_add__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1791 (all P_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(sRStar,P_5))))) # label(fact_3362_SRStar__finite) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1792 (all B_1_1 all B_2_1 all B_3_1 is_bool(tendsto_nat_real(B_1_1,B_2_1,B_3_1))) # label(gsy_c_Limits_Otendsto_000tc__Nat__Onat_000tc__RealDef__Oreal) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1793 (all Z_1 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),Z_1)),Z_1) != zero_zero_int) # label(fact_719_odd__nonzero) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1794 (all N_1 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_1)))))) # label(fact_3548_not__zle__0__negative) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1795 (all Xa_3 all X_16 all Y_12 hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Xa_3),X_16)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Xa_3),Y_12)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Xa_3),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_16),Y_12))) # label(fact_2489_mult__right_Odiff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1796 (all B_2 all A_3 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)),zero_zero_int),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),P_3)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(B_2),zero_zero_int),P_3)))))) # label(fact_910_zcong__zprime__prod__zero) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1797 (all A_235 all B_164 all C_95 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_235),B_164)),C_95) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_235),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_164),C_95))) # label(fact_220_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1798 (all Y_1 all X_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_1)) -> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),hAPP_real_real(hAPP_n546711566l_real(root,N_1),Y_1)) = hAPP_real_real(hAPP_n546711566l_real(root,N_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),Y_1)))))) # label(fact_3930_real__root__mult__lemma) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1799 (all B_96 all A_117 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_117),zero_z126310315umeral)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_96),zero_z126310315umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_117),B_96)),zero_z126310315umeral))))) # label(fact_1862_add__neg__nonpos) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1800 (all V_23 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_23)),one_one_int) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_23),hAPP_int_int(bit1,pls)))) # label(fact_54_add__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1801 (all M all N_1 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) -> M = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)))) # label(fact_1663_add__diff__inverse) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1802 (all A_184 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_184),zero_zero_rat))) # label(fact_915_dvd__0__right) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1803 (all N_58 ((N_58 != zero_zero_nat -> zero_zero_complex = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,zero_zero_complex),N_58)) & (zero_zero_nat = N_58 -> one_one_complex = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,zero_zero_complex),N_58)))) # label(fact_598_power__0__left) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1804 (all B_2 all Bq all Br all A_3 all C_1 all Aq all Ar (hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(A_3,C_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Aq),Ar))) -> (hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(B_2,C_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Bq),Br))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_1)) -> hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_3),B_2),C_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Aq),Bq)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ar),Br)),C_1))),hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ar),Br)),C_1)))))))) # label(fact_4213_divmod__nat__rel__add1__eq) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1805 (all A_192 all M_20 all N_43 hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_192),M_20)),N_43) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_192),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_20),N_43))) # label(fact_833_power__mult) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1806 (all N_1 all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),M)) -> N_1 != M)) # label(fact_1687_less__not__refl2) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1807 (all X_1 archim791455193or_rat(X_1) = archim1246769320r_real(ratreal(X_1))) # label(fact_4681_real__floor__code) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1808 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)),zero_zero_int)) -> hAPP_int_int(div_mod_int(A_3),B_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)))) # label(fact_2458_mod__pos__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1809 (all N hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),N)))))) # label(fact_4736_bdd__int__set__l__finite) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1810 (all V_16 all W_14 all Z_14 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_16)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,W_14)),Z_14)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_16),W_14))),Z_14)) # label(fact_423_mult__number__of__left) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1811 (all X_21 all N_31 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31)) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_21),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_31),one_one_nat))),X_21) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_21),N_31))) # label(fact_1736_realpow__minus__mult) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1812 (all P all Q all R hAPP_b1482221219l_real(P,hAPP_real_bool(Q,R)) = hAPP_r931665397l_real(hAPP_f1607917947l_real(hAPP_f1076489843l_real(cOMBB_415306361l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__prod_Itc__RealDef__Oreal_) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1813 (all V_14 all W_12 hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_14),W_12)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_14)),hAPP_int_complex(number528085621omplex,W_12))) # label(fact_431_number__of__mult) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1814 (all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),B))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),zero_zero_real)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B),zero_zero_real)))) # label(fact_1249_zero__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1815 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),X_1))),hAPP_real_real(uminus_uminus_real,X_1)))))) # label(fact_3497_ln__one__minus__pos__upper__bound) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1816 (all X_2 all Y_2 all Z_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_2),Z_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_2),Z_2))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Y_2))))) # label(fact_1601_real__mult__le__cancel__iff1) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1817 (all C all D all A all B all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,D),B)),Ma)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,D),A)),Ma))))) # label(fact_1163_zcong__zmult__prop2) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1818 (all A all B all C all D all X_2 (is_int(C) & is_int(B) & is_int(A) -> (hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,D),hAPP_int_nat(nat,X_2)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),B)),C) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_int_nat(nat,X_2))) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zOdd)) -> (hAPP_int_int(legacy_zgcd(A),B) = one_one_int -> (one_one_int = hAPP_int_int(legacy_zgcd(B),C) -> (hAPP_int_int(legacy_zgcd(C),A) = one_one_int -> (exists K exists L_1 exists M_1 (is_int(K) & hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K),hAPP_int_nat(nat,X_2)) = A & B = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,L_1),hAPP_int_nat(nat,X_2)) & hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,M_1),hAPP_int_nat(nat,X_2)) = C & is_int(M_1) & is_int(L_1))))))))))) # label(fact_4535_int__triple__relprime__odd__power__divisors) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1819 (all N_1 all X_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,N_1)),hAPP_int_real(real_int,X_1))),hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N_1),X_1)))))) # label(fact_4463_real__of__int__div2) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1820 (all A_119 all B_98 all V_6 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_119),B_98)),hAPP_int_complex(number528085621omplex,V_6)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_119),hAPP_int_complex(number528085621omplex,V_6))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_98),hAPP_int_complex(number528085621omplex,V_6)))) # label(fact_1561_left__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1821 (all V_7 all B_99 all C_55 hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_7)),B_99)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_7)),C_55)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_7)),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,B_99),C_55))) # label(fact_1557_right__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1822 (all P all Q all R hAPP_nat_nat(hAPP_nat_fun_nat_nat(P,R),hAPP_nat_nat(Q,R)) = hAPP_nat_nat(hAPP_f1914919701at_nat(hAPP_f1408247010at_nat(cOMBS_nat_nat_nat,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__Nat__Onat_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1823 (all I all J_2 all K_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J_2)),K_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),K_2)))) # label(fact_1758_add__lessD1) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1824 (all A_8 all N_14 all B_7 (is_int(B_7) & is_int(A_8) -> (hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_8),hAPP_nat_nat(suc,N_14)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_7),hAPP_nat_nat(suc,N_14)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_8)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_7)) -> B_7 = A_8))))) # label(fact_3117_power__inject__base) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1825 (all D_3 all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),A_3)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),B_2))))) # label(fact_4026_coprime__divprod) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1826 (all X_1 all Y_1 all Xa_2 all Ya hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Ya),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))))) # label(fact_3307_real__sqrt__sum__squares__mult__ge__zero) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1827 (all M all N_1 (hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N_1)) = one_one_int -> hAPP_int_int(abs_abs_int,M) = one_one_int)) # label(fact_3420_abs__zmult__eq__1) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1828 (all P_3 all R_3 all S_1 all Q_2 (is_int(Q_2) & is_int(S_1) -> (zero_zero_int != Q_2 -> (S_1 != zero_zero_int -> (normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,R_3),S_1)) = normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2)) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_3),Q_2) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,P_3),S_1)))))) # label(fact_4661_normalize__crossproduct) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1829 (all X_2 all Y_2 (hAPP_nat_int(semiri1621563631at_int,Y_2) = hAPP_nat_int(semiri1621563631at_int,X_2) <-> Y_2 = X_2)) # label(fact_3255_Nat__Transfer_Otransfer__int__nat__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1830 (all A_3 all B_2 all C_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_3),B_2)),C_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),C_1))) # label(fact_2718_div__mult2__eq) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1831 (all Ta all D all D_1 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),D_1)) -> (all X all K (-hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,K),D_1))),Ta))) <-> -hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X),Ta))))))) # label(fact_1819_inf__period_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1832 (all A_59 all B_51 all C_22 hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_59),B_51)),C_22) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_59),hAPP_nat_nat(div_mod_nat(B_51),C_22))),C_22)) # label(fact_2363_zmod__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1833 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3)) -> hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),A_3)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(div_mod_int(B_2),A_3))))) # label(fact_2498_pos__zmod__mult__2) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1834 (all P all Q all R hAPP_f1888375463t_real(hAPP_i447943416t_real(P,R),Q) = hAPP_i157317923t_real(hAPP_f1613846310t_real(hAPP_f1720068861t_real(cOMBC_295337211t_real,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Nat__Onat_Mtc__RealDef__Or) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1835 (all W_1 all Z_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),Z_2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_2),one_one_int))))) # label(fact_133_zle__add1__eq__le) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1836 (all A_222 A_222 = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_222),hAPP_int_rat(number_number_of_rat,pls))) # label(fact_420_add__numeral__0__right) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1837 (all M all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N_1)) # label(fact_3076_diff__Suc__eq__diff__pred) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1838 (all K_1 (is_int(K_1) -> (hAPP_int_int(bit0,K_1) = pls <-> pls = K_1))) # label(fact_352_rel__simps_I44_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1839 (all D all Sa exists Z all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z),X)) -> (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Sa))) <-> -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Sa)))))) # label(fact_2258_pinf_I10_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1840 (all N_1 all M all R_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),R_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N_1),R_3)) -> (hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N_1) = one_one_int -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N_1)),R_3)))))) # label(fact_4963_divides__mult__int) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1841 (all A_165 all B_140 hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_165),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_165),B_140)))) # label(fact_1136_dvd__triv__left) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1842 (all X_1 hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_1)) = hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_3738_minus__sin__cos__eq) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1843 (all N_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(fact,N_1)))) # label(fact_4563_fact__lt) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1844 (all X_1 all A_3 archim1246769320r_real(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),hAPP_int_real(real_int,A_3))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,archim1246769320r_real(X_1)),A_3)) # label(fact_4425_floor__subtract) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1845 (all Y_2 all X_2 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven)) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_2),Y_2)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_2),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),Y_2))) # label(fact_3237_even__prod__div__2) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1846 (all Q_2 all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (hBOOL(hAPP_nat_bool(prime,Q_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),Q_2)) -> P_3 = Q_2)))) # label(fact_4164_primes__eq) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1847 (all A all B ((all D_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),A)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),B)) <-> D_2 = hAPP_nat_nat(suc,zero_zero_nat))) <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A),B) = one_one_nat)) # label(fact_4371_coprime__Suc__0__nat) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1848 (all A_102 all C_45 all B_85 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_102),C_45)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_85),C_45))) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_102),B_85)))) # label(fact_2000_add__le__imp__le__right) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1849 (all A_187 all N_37 all N_36 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_37),N_36)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_187)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_187),N_37)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_187),N_36)))))) # label(fact_891_power__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1850 (all A all B all P_5 (hBOOL(hAPP_nat_bool(prime,P_5)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_5),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A),B))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_5),A)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_5),B))))) # label(fact_4168_prime__divprod__eq) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1851 (all M all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1))) # label(fact_4331_gcd__add2__nat) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1852 (all N_1 all M all R_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,M)),R_3)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N_1)),one_one_real))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,M),N_1))))) # label(fact_4462_lemma__floor) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1853 (all M all N_1 all R_5 all R_3 all S_3 all S_1 all T_2 all T_1 all K_2 (is_int(K_2) -> ((exists Sn exists Tn xzgcda(M,N_1,R_5,R_3,S_3,S_1,T_2,T_1) = hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,K_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Sn),Tn))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),R_3)) -> hAPP_int_int(legacy_zgcd(R_5),R_3) = K_2)))) # label(fact_4531_xzgcd__correct__aux2) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1854 (all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)))) # label(fact_4194_prime__ge__2) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1855 (all X_1 all Y_1 hAPP_int_int(div_mod_int(hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_1)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(div_mod_nat(X_1),Y_1))) # label(fact_3088_Divides_Otransfer__int__nat__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1856 (all Ma all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),N)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),Ma)))) # label(fact_3138_one__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1857 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,natfloor(X_1))),X_1)))) # label(fact_3673_real__natfloor__le) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1858 (all A_1 (hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1)) -> A_1 = image_nat_int(semiri1621563631at_int,image_int_nat(nat,A_1)))) # label(fact_4617_transfer__nat__int__set__return__embed) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1859 (all F all A_1 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma)) -> (hBOOL(hAPP_f448129468l_bool(resSet(Ma),A_1)) -> ((all X (is_int(X) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X),A_1)) -> (all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),A_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(F,X)),hAPP_int_int(F,Xa)),Ma)) -> X = Xa))))))) -> hBOOL(hAPP_f448129468l_bool(resSet(Ma),image_int_int(F,A_1))))))) # label(fact_4598_ResSet__image) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1860 (all F (hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,abs_abs_real),F),zero_zero_real,sequentially)) <-> hBOOL(tendsto_nat_real(F,zero_zero_real,sequentially)))) # label(fact_5041_LIMSEQ__rabs__zero) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1861 (all Lx_6 all Ly_4 all Rx_6 all Ry_4 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_6),Ly_4)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_6),Ry_4)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_6),Rx_6)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ly_4),Ry_4))) # label(fact_139_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1862 (all K_2 hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,K_2)),one_one_int) = quotient_of(hAPP_int_rat(number_number_of_rat,K_2))) # label(fact_4663_quotient__of__number_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.47 1863 (all X_32 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_32),zero_zero_nat) = one_one_int) # label(fact_948_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1864 (all P all Q all R hAPP_rat_bool(hAPP_i16292313t_bool(P,R),Q) = hAPP_int_bool(hAPP_r1900510169t_bool(hAPP_f1480506641t_bool(cOMBC_int_rat_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Rat__Orat_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1865 (all A all B (B = A <-> quotient_of(A) = quotient_of(B))) # label(fact_4654_quotient__of__inject__eq) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1866 (all C_80 all A_164 all B_139 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_164),B_139)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_164),C_80)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_164),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_139),C_80)))))) # label(fact_1142_dvd__add) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1867 (all N all X_2 hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,power_power_real),N),X_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,zero_zero_nat))))))) # label(fact_4780_DERIV__pow) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1868 (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2)) <-> X_2 = hAPP_real_real(exp_real,hAPP_real_real(ln,X_2)))) # label(fact_3847_exp__ln__iff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1869 (all Y_1 all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),Y_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_int_nat(nat,X_1)),hAPP_int_nat(nat,Y_1)))) # label(fact_2720_nat__div__distrib) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1870 (all X_1 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_486_quartic__square__square) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1871 (all X_1 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> hAPP_int_int(fact_fact_int,X_1) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(fact_fact_nat,hAPP_int_nat(nat,X_1)))) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> zero_zero_int = hAPP_int_int(fact_fact_int,X_1)))) # label(fact_4270_fact__int__def) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1872 (all P_3 all M ((M = zero_zero_nat -> one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),M)) & (M != zero_zero_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),M)))) # label(fact_1732_power__eq__if) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1873 (all N_66 all A_217 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_217)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_217),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_217),N_66)))))) # label(fact_525_power__gt1__lemma) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1874 (all A all B (zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B) <-> B = A)) # label(fact_2087_eq__iff__diff__eq__0) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1875 (all X_2 hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),sin)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(cos,X_2))),hAPP_real_real(sin,X_2))))) # label(fact_4793_DERIV__sin__realpow2a) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1876 (all U hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(ord_at641636577an_int(zero_zero_int),U)))) # label(fact_4917_finite__atLeastZeroLessThan__int) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1877 (all A_89 A_89 = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,one_on1684967323de_int),A_89)) # label(fact_2119_mult__1) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1878 (all B_1_1 is_int(quickcheck_int_of(B_1_1))) # label(gsy_c_Quickcheck__Narrowing_Oint__of) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1879 (all X_1 all A_3 all B_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_3),B_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),B_2)))))) # label(fact_4406_powr__mono) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1880 (all X_1 all Y_1 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_complex_real(re,X_1)),hAPP_complex_real(re,Y_1)) = hAPP_complex_real(re,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_1),Y_1))) # label(fact_4121_Re_Oadd) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1881 (all A_233 all C_93 all D_18 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_233),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_93),D_18)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_93),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_233),D_18))) # label(fact_235_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1882 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(exp_real,X_1))))) # label(fact_3843_exp__gt__one) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1883 (all X_2 all Y_2 (zero_z891286103de_int = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_i1732201573de_int(number1226105091de_int,Y_2)),hAPP_i1732201573de_int(number1226105091de_int,X_2)) <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_i1732201573de_int(number1226105091de_int,X_2)),hAPP_i1732201573de_int(number1226105091de_int,Y_2))))) # label(fact_2442_dvd__eq__mod__eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1884 (all X_2 all Y_2 all F ((all X hBOOL(hAPP_real_bool(deriv_real(F,X),zero_zero_real))) -> hAPP_real_real(F,X_2) = hAPP_real_real(F,Y_2))) # label(fact_3950_DERIV__isconst__all) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1885 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),t)) -> (exists X exists Y (is_int(X) & is_int(Y) & hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))) # label(fact_2__0961_A_060_At_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_06) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1886 (all A_60 all B_52 all C_23 hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_60),B_52)),C_23) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(div_mod_nat(A_60),C_23)),hAPP_nat_nat(div_mod_nat(B_52),C_23))),C_23)) # label(fact_2359_mod__add__eq) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1887 (all A_226 all B_161 all V_18 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_226),B_161)),hAPP_int_rat(number_number_of_rat,V_18)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_226),hAPP_int_rat(number_number_of_rat,V_18))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_161),hAPP_int_rat(number_number_of_rat,V_18)))) # label(fact_384_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1888 (all X_2 hBOOL(hAPP_real_bool(isCont_real_real(sqrt),X_2))) # label(fact_5162_isCont__real__sqrt) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1889 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))))) # label(fact_2284_dvd__pos__nat) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1890 (all Y_2 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_n546711566l_real(root,N),Y_2)))))) # label(fact_3920_real__root__gt__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1891 (all A all B all C all D (hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C),D) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C),D))))) # label(fact_2097_diff__eq__diff__less) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1892 (all P all Q all R hAPP_r653725362t_bool(hAPP_f1927674023t_bool(hAPP_f1261108921t_bool(cOMBB_172850899l_real,P),Q),R) = hAPP_f729588221t_bool(P,hAPP_r1562208522t_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__fun_Itc__Nat__Onat_) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1893 (all X_20 all Y_16 all A_77 all B_66 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_20),A_77)),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,Y_16),B_66))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_20),A_77)),B_66))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_77),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,Y_16),B_66))) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_20),Y_16)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_77),B_66))) # label(fact_2290_mult_Oprod__diff__prod) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1894 (all M_19 all N_38 all A_188 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_188)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_188),M_19)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_188),N_38))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_19),N_38))))) # label(fact_887_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1895 (all A (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),A))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A)))) # label(fact_2200_zero__less__double__add__iff__zero__less__single__add) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1896 (all L_2 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,min),hAPP_int_int(bit0,L_2)) = hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,min),L_2))) # label(fact_1588_diff__bin__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1897 (all X_2 hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),cos_coeff)),hAPP_r474017924t_real(power_power_real,X_2))),hAPP_real_real(cos,X_2)))) # label(fact_4864_cos__converges) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1898 (all N_1 all D_3 all A_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),A_3)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_3),N_1))))) # label(fact_4017_coprime__exp) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1899 (all X_1 hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(fact_fact_nat,X_1)) = hAPP_int_int(fact_fact_int,hAPP_nat_int(semiri1621563631at_int,X_1))) # label(fact_4279_transfer__int__nat__factorial) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1900 (all V_2 hAPP_int_nat(number_number_of_nat,V_2) = hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,V_2))) # label(fact_2730_nat__number__of__def) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1901 (all X_1 all Y_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Y_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y_1),X_1)) # label(fact_4305_gcd__semilattice__nat_Oinf__commute) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1902 (all X_1 -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(abs_abs_real,X_1)),one_one_real)),X_1))) # label(fact_3394_abs__add__one__not__less__self) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1903 (all X_35 all Y_27 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_35),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_27),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_845_sum__power2__ge__zero) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1904 (all M all N_1 hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),M)),N_1) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,M)))),N_1)) # label(fact_3210_Suc__mod__eq__add3__mod) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1905 (all A_3 all B_2 all C_1 (is_int(A_3) -> (hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),B_2) = C_1 -> A_3 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_1),B_2)))) # label(fact_1536_Int2_Oaux1) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1906 (all Y_6 all P_6 all N_10 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_6),N_10)) -> hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N_10)),P_6)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),P_6))),Y_6))) # label(fact_3176_lemma__realpow__diff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1907 (all X_2 all Y_2 (zero_zero_real != X_2 | zero_zero_real != Y_2 <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_2),X_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_2),Y_2)))))) # label(fact_739_sum__squares__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1908 (all Xa_3 all X_16 all Y_12 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa_3),X_16)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa_3),Y_12)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa_3),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_16),Y_12))) # label(fact_2488_mult__right_Odiff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1909 (all X hAPP_real_real(sin,X) = hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),sin_coeff)),hAPP_r474017924t_real(power_power_real,X)))) # label(fact_4819_lemma__sin__ext) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1910 (all Y_1 all X_1 all Z_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Z_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y_1),Z_1))))) # label(fact_4157_termination__basic__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1911 (all X_1 all Y_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_1)) -> (hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),N_1) = X_1 -> Y_1 = hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1))))) # label(fact_3932_real__root__pos__unique) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1912 (all P all Q all R hAPP_f252743847l_bool(P,hAPP_r373117745t_bool(Q,R)) = hAPP_r285062775l_bool(hAPP_f834119977l_bool(hAPP_f848516149l_bool(cOMBB_270132509l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Nat__Onat_Mtc__HOL__035) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1913 (all K_2 all I all J_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),J_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,K_2)),I)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,K_2)),J_2)))))) # label(fact_3155_zmult__zless__mono2__lemma) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1914 (all B_12 all A_15 all C_5 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_15),C_5) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_12),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_15),B_12))),hAPP_int_int(div_mod_int(A_15),B_12))),C_5)) # label(fact_2679_div__mod__equality2) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1915 (all A_165 all B_140 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_165),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_165),B_140)))) # label(fact_1140_dvd__triv__left) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1916 (all R_3 all A_3 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_3)),cis(A_3)) = rcis(R_3,A_3)) # label(fact_4265_rcis__def) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1917 (all N_1 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N_1)))) # label(fact_2990_zero__zle__int) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1918 (all A_46 all B_39 all C_14 hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_46),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_39),C_14))),B_39) = hAPP_Q1762011733de_int(div_mo231679042de_int(A_46),B_39)) # label(fact_2422_mod__mult__self2) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1919 (all B_1_1 all B_2_1 is_bool(hAPP_f215623910l_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__HOL__Obool_) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1920 (all A_3 all B_2 cis(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_3),B_2)) = hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,cis(A_3)),cis(B_2))) # label(fact_4150_cis__divide) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1921 (all C_29 all A_69 all B_61 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_29),hAPP_C498520661umeral(div_mo1740067990umeral(A_69),B_61)) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_29),A_69)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_29),B_61))) # label(fact_2328_mod__mult__mult1) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1922 (all A (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int)))) # label(fact_687_even__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1923 (all W_5 ((hAPP_int_nat(number_number_of_nat,W_5) != zero_zero_nat -> hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,zero_z126310315umeral),hAPP_int_nat(number_number_of_nat,W_5)) = zero_z126310315umeral) & (zero_zero_nat = hAPP_int_nat(number_number_of_nat,W_5) -> hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,zero_z126310315umeral),hAPP_int_nat(number_number_of_nat,W_5)) = one_on1645066479umeral))) # label(fact_1482_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1924 (all P all Q all R hAPP_n546711566l_real(hAPP_r195310020l_real(hAPP_f2126667875l_real(cOMBC_746811033l_real,P),Q),R) = hAPP_r1250527377l_real(hAPP_n682674106l_real(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__fun_Itc__Real) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1925 (all X_1 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim856651990g_real(X_1)),A_3)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_int_real(real_int,A_3))))) # label(fact_4421_ceiling__le__real) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1926 (all P all Q (-hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q)) | hBOOL(P))) # label(help_fconj_2_1_U) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1927 (all X_2 all N hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(sin,X_2)),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_r195310020l_real(hAPP_f2126667875l_real(cOMBC_746811033l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),zero_zero_real)),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,div_div_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,minus_minus_nat),hAPP_nat_nat(suc,zero_zero_nat)))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))))),hAPP_r474017924t_real(power_power_real,X_2))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(abs_abs_real,X_2)),N))))) # label(fact_4893_Maclaurin__sin__bound) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1928 (all P_1 all A all B (hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,A),B))) <-> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),B)) -> hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))) & (all D_2 (A = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B),D_2) -> hBOOL(hAPP_nat_bool(P_1,D_2)))))) # label(fact_1682_nat__diff__split) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1929 (all Q all P (hBOOL(hAPP_bool_bool(fimplies(P),Q)) | hBOOL(P))) # label(help_fimplies_1_1_U) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1930 (all G all X_2 all Ma (hBOOL(hAPP_real_bool(deriv_real(G,X_2),Ma)) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,cos),G),X_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,hAPP_real_real(G,X_2)))),Ma))))) # label(fact_4760_DERIV__fun__cos) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1931 (all A_68 all C_28 all B_60 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_68),B_60)),C_28) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_68),C_28)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_60),C_28))) # label(fact_2332_mod__mult__mult2) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1932 (all M_7 all N_21 hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_7),N_21)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(semiri132038758t_real,M_7)),hAPP_nat_real(semiri132038758t_real,N_21))) # label(fact_2966_of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1933 (all Z_1 (is_int(Z_1) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),one_one_int) = Z_1)) # label(fact_361_zmult__1__right) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1934 (all I all J_2 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(legacy_zgcd(I),J_2)),I))) # label(fact_4482_zgcd__zdvd1) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1935 (all C_83 all D_16 all A_169 all B_144 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_169),B_144)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,C_83),D_16)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_169),C_83)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_144),D_16)))))) # label(fact_1110_mult__dvd__mono) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1936 (all X_2 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,X_2)),hAPP_real_real(sqrt,Y_2))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Y_2)))) # label(fact_3317_real__sqrt__less__iff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1937 (all A_8 all N_14 all B_7 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_8),hAPP_nat_nat(suc,N_14)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_7),hAPP_nat_nat(suc,N_14)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_8)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_7)) -> B_7 = A_8)))) # label(fact_3116_power__inject__base) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1938 (all A all W_1 (zero_z891286103de_int = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A),hAPP_int_nat(number_number_of_nat,W_1)) <-> zero_zero_nat != hAPP_int_nat(number_number_of_nat,W_1) & A = zero_z891286103de_int)) # label(fact_1387_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1939 (all C_42 all A_99 all B_82 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_99),B_82)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_99),C_42)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_82),C_42))))) # label(fact_2018_add__right__mono) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1940 (all W pls = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,pls),W)) # label(fact_355_mult__Pls) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1941 (all X_1 all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> (one_one_real != A_3 -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> X_1 = hAPP_real_real(hAPP_r1250527377l_real(powr,A_3),hAPP_real_real(log(A_3),X_1)))))) # label(fact_4461_powr__log__cancel) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1942 (all K_1 all Ma all N (hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),N) <-> N = Ma)) # label(fact_1720_nat__add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1943 (all X_1 all Y_1 (hAPP_nat_nat(suc,Y_1) = hAPP_nat_nat(suc,X_1) -> X_1 = Y_1)) # label(fact_2829_Suc__inject) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1944 (all M all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1)),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N_1)) # label(fact_4332_gcd__add1__nat) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1945 (all R_3 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),R_3)) -> -(all S_2 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),S_2)) -> (all T (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),T)) -> R_3 != hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,S_2),T))))))) # label(fact_4664_obtain__pos__sum) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1946 (all X_19 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_19),zero_zero_real) = zero_zero_real) # label(fact_2476_mult__right_Ozero) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1947 (all N_1 -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_1))) # label(fact_1686_less__irrefl__nat) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1948 (all Y_1 all X_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),hAPP_real_real(hAPP_n546711566l_real(root,N_1),Y_1))))))) # label(fact_3928_real__root__less__mono__lemma) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1949 (all Q_2 ((hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Q_2),zero_zero_rat)) -> hAPP_rat_rat(abs_abs_rat,Q_2) = hAPP_rat_rat(uminus_uminus_rat,Q_2)) & (-hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Q_2),zero_zero_rat)) -> Q_2 = hAPP_rat_rat(abs_abs_rat,Q_2)))) # label(fact_4648_abs__rat__def) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1950 (all Y_1 all X_1 all N_1 (-hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1)) -> (X_1 = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),N_1) -> hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1) = Y_1))) # label(fact_3900_odd__real__root__unique) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1951 (all A_3 zero_zero_int = hAPP_int_int(div_mod_int(A_3),A_3)) # label(fact_2394_zmod__self) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1952 (all X_3 (hBOOL(summable_complex(X_3)) -> hAPP_complex_complex(cnj,hAPP_f213450978omplex(suminf_complex,X_3)) = hAPP_f213450978omplex(suminf_complex,hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,cnj),X_3)))) # label(fact_4843_cnj_Osuminf) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1953 (all K_1 all L (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),L)) <-> (exists N_2 L = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),N_2)))) # label(fact_3370_le__Suc__ex__iff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1954 (all X_2 (hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_nat_int(semiri1621563631at_int,X_2))) <-> hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2)))) # label(fact_3765_even__nat__def) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1955 (all N_25 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_nat_rat(semiri151668891at_rat,N_25)))) # label(fact_2906_of__nat__0__le__iff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1956 (all A (one_one_nat = A <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),one_one_nat)))) # label(fact_2280_gcd__lcm__complete__lattice__nat_Obot__unique) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1957 (all X_2 hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),sin)),sin),X_2),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_2)),hAPP_real_real(sin,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_2)),hAPP_real_real(sin,X_2)))))) # label(fact_4741_DERIV__sin__sin__mult) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1958 (all A_112 all B_91 all C_51 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_112),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_91),C_51)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_112),B_91)),C_51)) # label(fact_1908_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1959 (all Ma all N hAPP_nat_nat(nat_case_nat(zero_zero_nat,hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,suc),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,ord_min_nat),N))),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,Ma),hAPP_nat_nat(suc,N))) # label(fact_5124_min__Suc2) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1960 (all P_1 all A0 all A1 (hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(lazy_small_lazy_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A0),A1))) -> ((all D_2 all I_2 (is_int(D_2) & is_int(I_2) -> (hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(lazy_small_lazy_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,D_2),I_2))) -> ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,D_2),I_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,D_2),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I_2),one_one_int)))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,D_2),I_2)))))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A0),A1))))) # label(fact_4573_small__lazy_H_Opinduct) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1961 (all Z_8 all X_25 all Y_19 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,X_25),Y_19)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,X_25),Z_8)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,X_25),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,Y_19),Z_8)))))) # label(fact_1530_dvd__diff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1962 (all A_68 all C_28 all B_60 hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_68),C_28)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_60),C_28)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_68),B_60)),C_28)) # label(fact_2333_mod__mult__mult2) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1963 (all Y_1 all U_1 all X1_1 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X1_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X1_1),Y_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),U_1))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_1),hAPP_real_real(inverse_inverse_real,X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,U_1),hAPP_real_real(inverse_inverse_real,X1_1)))))))) # label(fact_3837_real__mult__inverse__cancel2) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1964 (all X_2 all Y_2 all Z_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_2),X_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_2),Y_2)))))) # label(fact_1602_real__mult__le__cancel__iff2) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1965 (all A all B all C (hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A),C) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A),B) <-> B = C)) # label(fact_1940_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1966 (all K_2 hAPP_nat_rat(semiri151668891at_rat,K_2) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_nat_int(semiri1621563631at_int,K_2)),one_one_int)) # label(fact_4675_of__nat__rat) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1967 (all C_78 all D_14 all A_159 all B_135 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_159),B_135)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_78),D_14)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_159)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_78)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_159),C_78)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_135),D_14)))))))) # label(fact_1193_mult__mono_H) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1968 (all A_193 all M_21 all N_45 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_193),M_21)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_193),N_45)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_193),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_21),N_45))) # label(fact_823_power__add) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1969 (all N_1 all X_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,N_1)),hAPP_nat_real(real_nat,X_1))),hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N_1),X_1)))))) # label(fact_3698_real__of__nat__div2) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1970 (all Y_1 all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_1)))))) # label(fact_5007_transfer__nat__int__gcd__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1971 (all A_29 all B_25 all C_10 (is_int(C_10) -> (C_10 != zero_zero_int -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_29),B_25) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_10),A_29)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_10),B_25))))) # label(fact_2581_div__mult__mult1) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1972 (all I_1 all M_2 (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,zero_zero_nat),M_2)) -> hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),M_2))),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),hAPP_nat_nat(suc,I_1))))) != zero_zero_nat)) # label(fact_4768_card__less) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1973 (all A_110 all B_89 all C_49 (hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_110),C_49) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_110),B_89) -> C_49 = B_89)) # label(fact_1923_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1974 (all X_1 all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(exp_real,X_1)),hAPP_real_real(exp_real,Y_1))))) # label(fact_3818_exp__less__mono) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1975 (all A_3 hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),zero_zero_int) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,zero_zero_int),one_one_int)) # label(fact_4669_eq__rat_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1976 (all A_3 all N_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_1)) -> (one_one_int = hAPP_int_int(legacy_zgcd(A_3),N_1) -> (exists X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),X)),one_one_int),N_1)) & is_int(X)))))) # label(fact_4533_zcong__lineq__ex) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1977 (all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),hAPP_int_int(bit0,K_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),K_1)))) # label(fact_939_rel__simps_I8_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1978 (all A_208 A_208 = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,zero_z891286103de_int),A_208)) # label(fact_627_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1979 (all X_1 all Y_1 hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),Y_1)) = hAPP_real_real(dist_dist_real(X_1),Y_1)) # label(fact_4544_dist__real__def) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1980 (all A_3 all B_2 all C_1 all Q_2 all R_3 (hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(B_2,C_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Q_2),R_3))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_1)) -> hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2),C_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),Q_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),R_3)),C_1))),hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),R_3)),C_1))))))) # label(fact_4212_divmod__nat__rel__mult1__eq) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1981 (all A_209 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_209),zero_zero_int) = zero_zero_int) # label(fact_615_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1982 (all Y_1 all X_1 all Z_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Z_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y_1),Z_1))))) # label(fact_4155_termination__basic__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1983 (all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> hAPP_i1948725293t_bool(d22set,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_i1948725293t_bool(wset(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),P_5))) # label(fact_2242_d22set__eq__wset) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1984 (all A_3 all B_2 hAPP_int_rat(ring_1_of_int_rat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(sgn_sgn_int,A_3)),hAPP_int_int(sgn_sgn_int,B_2))) = hAPP_rat_rat(sgn_sgn_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2))) # label(fact_4714_sgn__rat) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1985 (all X_2 all Y_2 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X_2)),hAPP_real_real(hAPP_n546711566l_real(root,N),Y_2))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Y_2))))) # label(fact_3902_real__root__le__iff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1986 (all A_3 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_3),B_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_2),A_3)) -> hAPP_int_int(abs_abs_int,A_3) = hAPP_int_int(abs_abs_int,B_2)))) # label(fact_3417_zdvd__antisym__abs) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1987 (all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),N_1) = N_1) # label(fact_5189_max__0L) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1988 (all B_1_1 all B_2_1 (is_int(B_2_1) -> is_int(hAPP_int_int(B_1_1,B_2_1)))) # label(gsy_c_hAPP_000tc__Int__Oint_000tc__Int__Oint) # label(hypothesis) # label(non_clause). [assumption]. 5.43/5.48 1989 (all A all C all B (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A),B)) <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A),C)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B),C))))) # label(fact_2076_add__less__cancel__right) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1990 (all V all W_1 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_i769753017umeral(number1443263063umeral,V)),hAPP_i769753017umeral(number1443263063umeral,W_1))) <-> -hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_i769753017umeral(number1443263063umeral,W_1)),hAPP_i769753017umeral(number1443263063umeral,V))))) # label(fact_84_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1991 (all V_2 all W hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,W)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,V_2))),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W)))) # label(fact_2560_zdiv__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1992 (all N_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N_1)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1)))) # label(fact_3710_two__realpow__gt) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1993 (all V_5 all W_3 all C_54 hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_5),W_3))),C_54) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_5)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_int_rat(number_number_of_rat,W_3)),C_54))) # label(fact_1584_add__number__of__diff1) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1994 (all N_1 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> hAPP_real_real(hAPP_n546711566l_real(root,hAPP_nat_nat(suc,N_1)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_nat_nat(suc,N_1))) = X_1)) # label(fact_3927_real__root__pos) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1995 (all P all Q all R hAPP_i1948725293t_bool(P,hAPP_P1175774780nt_int(Q,R)) = hAPP_P252494982t_bool(hAPP_f291087797t_bool(hAPP_f214467655t_bool(cOMBB_182850971nt_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool__014) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1996 (all A all B (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),B)),zOdd)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),zOdd)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),zEven))))) # label(fact_3095_even__plus__odd__prop1) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1997 (all C_76 all B_133 all A_157 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_133),A_157)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_76),zero_zero_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_157),C_76)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_133),C_76)))))) # label(fact_1199_mult__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1998 (all P all X_1 (-hBOOL(hAPP_int_bool(P,X_1)) | hBOOL(hAPP_f448129468l_bool(fEx_int,P)))) # label(help_fEx_1_1_fEx_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 1999 (all Y_2 all X_2 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zOdd)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zOdd)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),Y_2)),zEven))))) # label(fact_3100_odd__minus__odd) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2000 (all A all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_2)) -> hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,A)),hAPP_r474017924t_real(power_power_real,X_2)),zero_zero_real,sequentially)))) # label(fact_5043_LIMSEQ__divide__realpow__zero) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2001 (all P all Q all R hAPP_r632163725t_bool(hAPP_f1832844496t_bool(hAPP_f1990636069t_bool(cOMBC_412863505t_bool,P),Q),R) = hAPP_f1048215610t_bool(hAPP_r1623423338t_bool(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Int__Oint_Mtc__HOL__O) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2002 (all Y_2 the_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real))),hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,ord_less_eq_real),pi))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),cos)),Y_2)))) = hAPP_real_real(arccos,Y_2)) # label(fact_4885_arccos__def) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2003 (all N_1 hAPP_nat_nat(fact,hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,N_1)),hAPP_nat_nat(fact,N_1))) # label(fact_4565_fact_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2004 (all Z_1 all X_1 all Y_1 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),Z_1)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_1),X_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Z_1))))) # label(fact_1715_dvd_Oless__le__trans) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2005 (all N_32 all M_18 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),M_18)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),N_32)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,M_18),N_32)))))) # label(fact_1365_less__1__mult) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2006 (all B_74 all A_85 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_85),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_74),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_85),B_74)),zero_zero_real))))) # label(fact_2151_add__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2007 (all N_60 all X_41 all Y_33 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,X_41),Y_33)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_41),N_60)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_33),N_60))))) # label(fact_583_dvd__power__same) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2008 (all X_32 one_on1684967323de_int = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_32),zero_zero_nat)) # label(fact_953_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2009 (all X_1 hAPP_complex_complex(cnj,hAPP_complex_complex(invers1449016382omplex,X_1)) = hAPP_complex_complex(invers1449016382omplex,hAPP_complex_complex(cnj,X_1))) # label(fact_4063_complex__cnj__inverse) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2010 (all M_25 all N_65 all A_216 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_216)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_216),M_25)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_216),N_65))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_25),N_65))))) # label(fact_531_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2011 (all Y_9 all Z_6 all X_13 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_6),X_13)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_6),Y_9)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X_13),Z_6)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,Y_9),Z_6)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_13),Y_9)),Z_6)))) # label(fact_2612_div__add) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2012 (all N_1 all M hAPP_i1732201573de_int(quickcheck_of_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,quickcheck_int_of(N_1)),quickcheck_int_of(M))) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,N_1),M)) # label(fact_4568_div__code__int__def) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2013 (all A_231 all N_68 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_231),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_68)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_231),N_68)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_279_power__even__eq) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2014 (all K_1 all Ma all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)))) # label(fact_1780_nat__add__left__cancel__le) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2015 (all X_45 all P_7 all Q_11 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_45),P_7)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_45),Q_11)) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_45),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_11))) # label(fact_98_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2016 (all A_229 all B_162 all C_91 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_229),B_162)),C_91) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_229),C_91)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_162),C_91))) # label(fact_308_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2017 (all X_9 all Q_7 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_9),hAPP_nat_nat(suc,Q_7)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_9),Q_7)),X_9)) # label(fact_3009_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2018 (all A_239 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_239),A_239) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_239),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_37_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2019 (all C all B all A (A != zero_z891286103de_int -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A),B)) -> (hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,B),A) = C <-> B = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C),A))))) # label(fact_2664_dvd__div__eq__mult) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2020 (all A all B all C all D (hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,C),D) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A),B) -> (B = A <-> C = D))) # label(fact_1961_diff__eq__diff__eq) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2021 (all B_44 all B_43 all A_52 all C_16 all A_51 (hAPP_int_int(div_mod_int(A_51),C_16) = hAPP_int_int(div_mod_int(A_52),C_16) -> (hAPP_int_int(div_mod_int(B_44),C_16) = hAPP_int_int(div_mod_int(B_43),C_16) -> hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_51),B_43)),C_16) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_52),B_44)),C_16)))) # label(fact_2377_mod__diff__cong) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2022 (all X_47 all P_8 all Q_12 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_47),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_8),Q_12)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_47),P_8)),Q_12)) # label(fact_73_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2023 (all N_1 -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),zero_zero_nat))) # label(fact_1742_less__nat__zero__code) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2024 (all Z1 all Z2 all W hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),Z2)),W) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),W)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z2),W))) # label(fact_364_zadd__zmult__distrib) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2025 (all B_71 all A_82 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_82)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_71)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_82),B_71)))))) # label(fact_2176_add__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2026 (all A (hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially)) -> (hBOOL(monoseq_real(A)) -> hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)))))) # label(fact_5048_summable__Leibniz_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2027 (all K_1 all Ma all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Ma),N))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),Ma)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),N)))) # label(fact_4985_gcd__greatest__iff__int) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2028 (all P all Q all R hAPP_i1216375074nt_int(hAPP_P621635040nt_int(hAPP_f1582586579nt_int(cOMBC_707974837nt_int,P),Q),R) = hAPP_P1145851913nt_int(hAPP_i2104874254nt_int(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2029 (all X_2 all Y_2 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_2),Y_2)),zEven)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven)) | hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven)))) # label(fact_2897_EvenOdd_Oeven__product) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2030 (all M all K_2 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K_2)),N_1)) -> -(hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N_1))))) # label(fact_1773_add__leE) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2031 (all A_135 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_135),A_135)),zero_zero_int))) # label(fact_1329_not__square__less__zero) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2032 (all N_58 ((zero_zero_nat = N_58 -> one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N_58)) & (zero_zero_nat != N_58 -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N_58) = zero_zero_nat))) # label(fact_595_power__0__left) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2033 (all E_1 all B_2 all D_3 all A_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),A_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,E_1),B_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),B_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),E_1)))))) # label(fact_4016_coprime__divisors) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2034 (all A_3 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),one_one_nat))) # label(fact_4013_coprime__1) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2035 (all P all Q all R hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(P,R),hAPP_i1948725293t_bool(Q,R)) = hAPP_i1948725293t_bool(hAPP_f428220345t_bool(hAPP_f1878066172t_bool(cOMBS_1720984575t_bool,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2036 (all A all B (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),B)),zOdd)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),zEven)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),zOdd))))) # label(fact_3094_even__plus__odd__prop2) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2037 (all K_2 all L_2 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(bit1,K_2)),hAPP_int_int(bit1,L_2)) = hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),L_2))) # label(fact_1564_diff__bin__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2038 (all X_2 hBOOL(hAPP_real_bool(deriv_real(cos,X_2),hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_2))))) # label(fact_3864_DERIV__cos) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2039 (all N_1 all M hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),M)))) # label(fact_1782_le__add1) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2040 (all A_3 hAPP_real_real(sin,A_3) = hAPP_complex_real(im,cis(A_3))) # label(fact_4146_Im__cis) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2041 (all P all Q all R hAPP_i68813070l_bool(hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,P),Q),R) = hAPP_b589554111l_bool(P,hAPP_int_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2042 (all A_33 all B_29 all C_12 (is_int(C_12) -> (zero_zero_int = C_12 -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_12),A_33)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_12),B_29)) = zero_zero_int) & (zero_zero_int != C_12 -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_12),A_33)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_12),B_29)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_33),B_29)))) # label(fact_2565_div__mult__mult1__if) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2043 (all K_2 all L_2 hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(abs_abs_int,K_2)),hAPP_int_int(abs_abs_int,L_2))),hAPP_int_int(div_mod_int(hAPP_int_int(abs_abs_int,K_2)),hAPP_int_int(abs_abs_int,L_2))) = pdivmod(K_2,L_2)) # label(fact_3945_pdivmod__def) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2044 (all A_3 all B_2 (A_3 != B_2 -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),A_3))))) # label(fact_1695_dvd_Oneq__le__trans) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2045 (all B_71 all A_82 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_82)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_71)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_82),B_71)))))) # label(fact_2175_add__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2046 (all A_207 A_207 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_207),zero_zero_nat)) # label(fact_631_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2047 (all A_105 A_105 = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,zero_zero_complex),A_105)) # label(fact_1984_add__0) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2048 (all N_1 N_1 = hAPP_int_nat(nat,hAPP_nat_int(semiri1621563631at_int,N_1))) # label(fact_2856_nat__int) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2049 (all B_97 all A_118 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_118),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_97),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_118),B_97)),zero_zero_real))))) # label(fact_1859_add__nonpos__neg) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2050 (all A all X_2 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),hAPP_int_nat(nat,X_2)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(uminus_uminus_int,A)),hAPP_int_nat(nat,X_2))))) # label(fact_3553_neg__even__power) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2051 (all A_202 all N_52 all N_51 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_52),N_51)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_202)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_202),one_on1684967323de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_202),N_51)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_202),N_52))))))) # label(fact_679_power__decreasing) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2052 (all N_1 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_nat_real(real_nat,N_1))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,natfloor(X_1)),N_1))))) # label(fact_3697_less__natfloor) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2053 (all X_2 all Y_2 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X_2)),hAPP_real_real(hAPP_n546711566l_real(root,N),Y_2)))))) # label(fact_3905_real__root__less__iff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2054 (all P all Q all R hAPP_i371647666l_real(hAPP_f1352333353l_real(hAPP_f318350629l_real(cOMBB_1946221070al_int,P),Q),R) = hAPP_r1250527377l_real(P,hAPP_int_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__R) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2055 (all Ma all N all K_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N))))) # label(fact_1854_nat__mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2056 (all M all N_1 (M = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1) -> N_1 = one_one_nat | M = zero_zero_nat)) # label(fact_1804_mult__eq__self__implies__10) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2057 (all P all Q all R hAPP_n1865633855at_nat(P,hAPP_nat_nat(Q,R)) = hAPP_n1865633855at_nat(hAPP_f837737749at_nat(hAPP_f999082675at_nat(cOMBB_348438404at_nat,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__prod_Itc__N) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2058 (all X_2 comple124823625p_real(hAPP_r1134773055l_bool(ord_lessThan_real,X_2)) = X_2) # label(fact_4956_Sup__lessThan) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2059 (all X_2 all Y_2 all Z_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y_2),Z_2))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Z_2)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2)))) # label(fact_4326_gcd__semilattice__nat_Ole__inf__iff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2060 (all X_2 all A_1 (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A_1)) <-> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_nat_int(semiri1621563631at_int,X_2)),image_nat_int(semiri1621563631at_int,A_1))))) # label(fact_4602_transfer__nat__int__set__relations_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2061 (all M all K_2 all N_1 (hAPP_int_int(legacy_zgcd(K_2),N_1) = one_one_int -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),N_1)),hAPP_int_int(legacy_zgcd(M),N_1))))) # label(fact_4516_zgcd__zmult__zdvd__zgcd) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2062 (all Sa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Sa)) -> hBOOL(tendsto_nat_real(hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,powr),real_nat)),hAPP_real_real(uminus_uminus_real,Sa)),zero_zero_real,sequentially)))) # label(fact_5046_LIMSEQ__neg__powr) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2063 (all B_150 all A_178 (A_178 != zero_zero_nat -> (zero_zero_nat != B_150 -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_178),B_150) != zero_zero_nat))) # label(fact_1026_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2064 (all D all F all X_2 all L (hBOOL(hAPP_real_bool(deriv_real(F,X_2),L)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D)) -> ((all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_2),Y))),D)) -> hAPP_real_real(F,X_2) = hAPP_real_real(F,Y))) -> zero_zero_real = L)))) # label(fact_3885_DERIV__local__const) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2065 (all M hAPP_nat_nat(div_mod_nat(M),hAPP_nat_nat(suc,zero_zero_nat)) = zero_zero_nat) # label(fact_3080_mod__1) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2066 (all D_1 all X_2 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (X_2 != zero_zero_real -> ((hBOOL(hAPP_nat_bool(even_odd_even_nat,N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2)) -> hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,zero_zero_nat))))) = D_1)) -> ((hBOOL(hAPP_nat_bool(even_odd_even_nat,N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),zero_zero_real)) -> hAPP_real_real(uminus_uminus_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,zero_zero_nat)))))) = D_1)) -> ((-hBOOL(hAPP_nat_bool(even_odd_even_nat,N)) -> D_1 = hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,zero_zero_nat)))))) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(root,N),X_2),D_1)))))))) # label(fact_3892_DERIV__real__root__generic) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2067 (all A_172 all B_147 all C_85 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_172),B_147)),C_85)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,B_147),C_85)))) # label(fact_1085_dvd__mult__right) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2068 (all M all N_1 hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M),N_1)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(semiri1621563631at_int,M)),N_1)) # label(fact_3083_int__power) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2069 (all A_208 A_208 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A_208)) # label(fact_624_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2070 (all W hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),W))) # label(fact_63_zle__refl) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2071 (all A_3 all B_2 all C_1 all D_3 hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_3),C_1)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_2),D_3)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),B_2)),hAPP_real_complex(hAPP_r265291036omplex(complex,C_1),D_3))) # label(fact_3963_complex__diff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2072 (all K_2 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),N_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N_1))))) # label(fact_1822_dvd__imp__le) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2073 (all X_7 all Q_5 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_7),hAPP_nat_nat(suc,Q_5)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_7),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_7),Q_5))) # label(fact_3020_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2074 (all M all D_3 all R_3 (R_3 = hAPP_nat_nat(div_mod_nat(M),D_3) -> (exists Q_4 M = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,R_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Q_4),D_3))))) # label(fact_2779_mod__eqD) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2075 (all I all J_2 all K_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),K_2)),J_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),J_2)),K_2)) # label(fact_1841_diff__commute) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2076 (all B_2 all A_3 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),B_2)) -> (exists D_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2)) & (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),Y))),D_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),B_2)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_3),Y))))))))) # label(fact_3411_lemma__interval) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2077 (all V_3 all V_2 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_int_nat(number_number_of_nat,V_3)) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_3),pls)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_2),V_3))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_3),pls)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_int_nat(number_number_of_nat,V_2))))) # label(fact_122_add__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2078 (all W all Z1 all Z2 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),Z2)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z2))) # label(fact_365_zadd__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 5.43/5.48 2079 (all A_177 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_177),zero_z891286103de_int) = zero_z891286103de_int) # label(fact_1041_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2080 (all M (is_int(M) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M)) -> hAPP_int_int(legacy_zgcd(M),M) = M))) # label(fact_4497_zgcd__self) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2081 (all Ma all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)))) # label(fact_3001_zle__int) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2082 (all C_39 all D_7 all A_96 all B_79 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_96),B_79)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_39),D_7)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_96),C_39)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_79),D_7)))))) # label(fact_2050_add__strict__mono) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2083 (all X_2 (-hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zOdd)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven)))) # label(fact_2902_not__odd__impl__even) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2084 (all B_128 all A_152 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_152),zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_128)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_152),B_128)),zero_zero_nat))))) # label(fact_1222_mult__nonpos__nonneg) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2085 (all M_27 all A_224 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_224),one_one_rat)),M_27) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,M_27),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_224),M_27))) # label(fact_405_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2086 (all X_23 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_23),X_23)),one_one_int) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_23),one_one_int)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_23),one_one_int))) # label(fact_1543_real__squared__diff__one__factored) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2087 (all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B)),zero_zero_real)))) # label(fact_2218_less__iff__diff__less__0) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2088 (all B all A (B = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B),A) <-> A = zero_zero_complex)) # label(fact_639_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2089 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B)),one_one_rat))))) # label(fact_4720_Fract__less__one__iff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2090 (all N_1 (N_1 != zero_zero_nat -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),one_one_nat)),N_1)))) # label(fact_4030_coprime__minus1) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2091 (all A_3 (hBOOL(hAPP_nat_bool(prime,A_3)) -> hBOOL(hAPP_int_bool(zprime,hAPP_nat_int(semiri1621563631at_int,A_3))))) # label(fact_4179_prime__impl__zprime__int) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2092 (all B_114 all A_137 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_137)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_114),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_137),B_114)),zero_zero_int))))) # label(fact_1308_mult__pos__neg) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2093 (all A_3 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))) -> hAPP_int_int(inv(P_3),A_3) != A_3)))) # label(fact_2128_inv__distinct) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2094 (all N_1 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),N_1) = hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1)))),pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3708_sin__cos__npi) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2095 (all P all Q all R hAPP_n546711566l_real(hAPP_f1904307534l_real(hAPP_f224989205l_real(cOMBB_1419615765al_nat,P),Q),R) = hAPP_n546711566l_real(P,hAPP_nat_nat(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__RealDef__Oreal_Mtc__RealDe) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2096 (all A all B (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A),B)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B)),zero_zero_rat)))) # label(fact_2216_le__iff__diff__le__0) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2097 (all A_124 all C_58 all B_103 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_124),C_58)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_103),C_58))) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_58)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_124),B_103))))) # label(fact_1460_mult__right__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2098 (all P all Q all R hAPP_r2115560837t_bool(P,hAPP_int_rat(Q,R)) = hAPP_i16292313t_bool(hAPP_f601898971t_bool(hAPP_f857296063t_bool(cOMBB_214503962ol_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Rat__Orat_000tc__fun_Itc__Rat__Orat_Mtc__HOL__Obool_) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2099 (all N_1 all M hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(div_mod_nat(M),N_1))) # label(fact_2746_mult__div__cancel) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2100 (all N all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Ma)) -> (one_one_nat = N <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),N)),Ma))))) # label(fact_1630_dvd__mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2101 (all X_1 all Y_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Y_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Y_1)) # label(fact_4307_gcd__semilattice__nat_Oinf__left__idem) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2102 (all L_2 hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,pls),L_2)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,pls),hAPP_int_int(bit0,L_2))) # label(fact_1565_diff__bin__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2103 (all Lx all Rx all Ry hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx),Ry)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx),Ry))) # label(fact_184_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2104 (all Ma all N (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(semiri1424489471de_int,Ma)),hAPP_n522471361de_int(semiri1424489471de_int,N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)))) # label(fact_2950_of__nat__le__iff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2105 (all P all Q all R hAPP_f1888375463t_real(P,hAPP_r474017924t_real(Q,R)) = hAPP_r474017924t_real(hAPP_f1993765077t_real(hAPP_f895854793t_real(cOMBB_1948233853l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_000tc__fun__017) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2106 (all C_88 all A_180 all B_152 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_180),B_152)) -> (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,B_152),C_88)) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_180),C_88))))) # label(fact_989_dvd__trans) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2107 (all A_3 all J_2 all K_2 all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(K_2),zero_zero_int),P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),zero_zero_int),P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),K_2))),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(K_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),J_2))),P_3)))))))) # label(fact_2222_MultInv__zcong__prop2) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2108 (all X_3 all A (hBOOL(hAPP_complex_bool(sums_complex(X_3),A)) -> hBOOL(hAPP_complex_bool(sums_complex(hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,cnj),X_3)),hAPP_complex_complex(cnj,A))))) # label(fact_4870_cnj_Osums) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2109 (all A_60 all B_52 all C_23 hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_60),C_23)),hAPP_C498520661umeral(div_mo1740067990umeral(B_52),C_23))),C_23) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_60),B_52)),C_23)) # label(fact_2360_mod__add__eq) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2110 (all N hAPP_n1699378549t_bool(ord_lessThan_nat,N) = hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N)) # label(fact_4951_atLeast0LessThan) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2111 (all M_26 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,M_26),M_26) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,one_on1684967323de_int),one_on1684967323de_int)),M_26)) # label(fact_411_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2112 (all Ma all N (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(semiri1619134803umeral,Ma)),hAPP_n1108039445umeral(semiri1619134803umeral,N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)))) # label(fact_2931_of__nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2113 (all P_1 all N ((exists X (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))) & hBOOL(hAPP_nat_bool(P_1,X)))) <-> (exists M_1 (hBOOL(hAPP_nat_bool(P_1,M_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N)))))) # label(fact_4900_ex__nat__less__eq) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2114 (all A_3 all N_1 all B_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_3),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_2),N_1))) # label(fact_4301_gcd__exp__nat) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2115 (all K_2 all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),K_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),K_2)),X_1))))) # label(fact_2621_int__div__less__self) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2116 (all N all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N)),hAPP_nat_nat(suc,Ma))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),Ma)))) # label(fact_2874_Suc__le__mono) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2117 (all M all K_2 all N_1 (is_int(K_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_1)) -> (K_2 = hAPP_int_int(legacy_zgcd(M),N_1) -> (exists S_2 exists T (is_int(S_2) & hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,S_2),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,T),N_1)) = K_2 & is_int(T))))))) # label(fact_4536_zgcd__ex__linear) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2118 (all C_46 all A_103 all B_86 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_46),A_103)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_46),B_86))) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_103),B_86)))) # label(fact_1994_add__le__imp__le__left) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2119 (all A_71 all B_63 all C_31 hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_71),B_63)),C_31) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_71),C_31)),B_63)),C_31)) # label(fact_2321_mod__mult__left__eq) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2120 (all C (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,C)),one_one_real)) -> hBOOL(tendsto_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(abs_abs_real,C)),zero_zero_real,sequentially)))) # label(fact_5038_LIMSEQ__rabs__realpow__zero) # label(axiom) # label(non_clause). [assumption]. 5.43/5.49 2121 (all N hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f451358433l_real(hAPP_f335634176l_real(cOMBS_337378985l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,div_div_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,zero_zero_real)))),zero_zero_real)),hAPP_n1699378549t_bool(ord_at4362885an_nat(one_one_nat),N)) = zero_zero_real) # label(fact_4911_lemma__STAR__cos2) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2122 (all W all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W),Z_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W),one_one_int)),Z_1)))) # label(fact_131_zless__imp__add1__zle) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2123 (all C_46 all A_103 all B_86 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_46),A_103)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_46),B_86))) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_103),B_86)))) # label(fact_1999_add__le__imp__le__left) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2124 (all A all B all C all D (hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),D) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C),D))))) # label(fact_2098_diff__eq__diff__less) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2125 (all B_1 all P_2 all P_1 all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1)) -> ((exists Z all X (is_int(X) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),Z)) -> (hBOOL(hAPP_int_bool(P_1,X)) <-> hBOOL(hAPP_int_bool(P_2,X)))))) -> ((all X (is_int(X) -> ((all Xa (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) -> (all Xb (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B_1)) -> X != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa))))) -> (hBOOL(hAPP_int_bool(P_1,X)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),D_1))))))) -> ((all X all K (is_int(X) & is_int(K) -> (hBOOL(hAPP_int_bool(P_2,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D_1)))) <-> hBOOL(hAPP_int_bool(P_2,X))))) -> ((exists X1 (hBOOL(hAPP_int_bool(P_1,X1)) & is_int(X1))) <-> (exists X (hBOOL(hAPP_int_bool(P_2,X)) & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) & is_int(X))) | (exists X (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) & (exists Xa (is_int(Xa) & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),B_1)) & hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xa),X))))) & is_int(X))))))))) # label(fact_5153_cpmi) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2126 (all X_1 (is_int(X_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) -> X_1 = zero_zero_int | X_1 = one_one_int)))) # label(fact_898_int__pos__lt__two__imp__zero__or__one) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2127 (all P all Q all R hAPP_n1699378549t_bool(hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,P),Q),R) = hAPP_n1699378549t_bool(P,hAPP_nat_nat(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2128 (all N_1 all X_1 all Y_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_1),hAPP_nat_nat(suc,N_1)))))) # label(fact_3090_divides__rexp) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2129 (all Z_1 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Z_1),hAPP_complex_complex(cnj,Z_1)) = of_real_complex(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_complex_real(re,Z_1)))) # label(fact_4136_complex__add__cnj) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2130 (all V_9 all U_2 all Y_20 all X_26 all A_122 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_26),A_122)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_20),A_122)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),U_2)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),V_9)) -> (hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,U_2),V_9) = one_one_rat -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,U_2),X_26)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,V_9),Y_20))),A_122)))))))) # label(fact_1492_convex__bound__le) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2131 (all A_198 A_198 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_198),one_one_nat)) # label(fact_751_power__one__right) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2132 (all C_70 all A_145 all B_122 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_145),B_122)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_70)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_70),A_145)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_70),B_122)))))) # label(fact_1264_comm__mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2133 (all Q_2 all R_3 hAPP_P1975530577nt_int(negateSnd,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),hAPP_int_int(uminus_uminus_int,R_3))) # label(fact_4188_negateSnd__eq) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2134 (all R_3 all X_1 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_complex_real(re,X_1)) = hAPP_complex_real(re,hAPP_complex_complex(scaleR1652505878omplex(R_3),X_1))) # label(fact_5108_complex__Re__scaleR) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2135 (all Y_1 all X_1 (zero_zero_real != hAPP_real_real(cos,X_1) -> (zero_zero_real != hAPP_real_real(cos,Y_1) -> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(tan,X_1)),hAPP_real_real(tan,Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_1)),hAPP_real_real(cos,Y_1)))))) # label(fact_3712_add__tan__eq) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2136 (all A_179 all B_151 (is_int(A_179) & is_int(B_151) -> (hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_179),B_151) = zero_zero_int -> B_151 = zero_zero_int | A_179 = zero_zero_int))) # label(fact_1018_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2137 (all M all N_1 all Qr_1 (hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N_1),Qr_1)) -> Qr_1 = divmod_nat(M,N_1))) # label(fact_4215_divmod__nat__eq) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2138 (all X_2 (X_2 = one_one_nat <-> X_2 = one_one_nat)) # label(fact_1953_one__reorient) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2139 (all X_2 all Y_2 all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),B)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),Y_2)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B),X_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B),Y_2)))))) # label(fact_879_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2140 (all N_66 all A_217 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_217)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_217),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_217),N_66)))))) # label(fact_527_power__gt1__lemma) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2141 (all Ma all W_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),W_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat,W_1)),Ma)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),hAPP_nat_int(semiri1621563631at_int,Ma)))))) # label(fact_3190_nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2142 (all X_1 all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,pi)),Y_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(cos,Y_1)),hAPP_real_real(cos,X_1))))))) # label(fact_3577_cos__monotone__minus__pi__0) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2143 (all P_1 all A all B (-(-hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),B)) | (exists D_2 (-hBOOL(hAPP_nat_bool(P_1,D_2)) & A = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B),D_2)))) <-> hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,A),B))))) # label(fact_1681_nat__diff__split__asm) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2144 (all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(suc,zero_zero_nat))))) # label(fact_4181_prime__nd__one) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2145 (all X_2 all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),Y_2)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(number_number_of_rat,X_2)),hAPP_int_rat(number_number_of_rat,Y_2))))) # label(fact_95_le__number__of) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2146 (all Z_1 hAPP_complex_real(norm_norm_complex,Z_1) = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,Z_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(im,Z_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_4137_cmod__def) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2147 (all X_2 all Y_2 (hAPP_nat_nat(div_mod_nat(hAPP_int_nat(number_number_of_nat,Y_2)),hAPP_int_nat(number_number_of_nat,X_2)) = zero_zero_nat <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_int_nat(number_number_of_nat,X_2)),hAPP_int_nat(number_number_of_nat,Y_2))))) # label(fact_2444_dvd__eq__mod__eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2148 (all B all P_5 all A (is_int(B) & is_int(A) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A)) -> (-hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(wset(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),one_one_int)),P_5))) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(wset(A),P_5))) -> B = A | B = hAPP_int_int(inv(P_5),A)))))) # label(fact_2240_wset__mem__imp__or) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2149 (all R_2 hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(plus_plus_real,R_2)),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),suc))),R_2,sequentially))) # label(fact_5039_LIMSEQ__inverse__real__of__nat__add) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2150 (all B_1_1 is_int(archim791455193or_rat(B_1_1))) # label(gsy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_000tc__Rat__Orat) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2151 (all X_1 hAPP_real_real(cos,hAPP_real_real(arctan,X_1)) != zero_zero_real) # label(fact_3563_cos__arctan__not__zero) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2152 (all C_77 all B_134 all A_158 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_134),A_158)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_77),zero_zero_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_77),A_158)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_77),B_134)))))) # label(fact_1196_mult__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2153 (all B_138 all A_162 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_162)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_138),zero_zero_int)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_138)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_162),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_162),B_138)),zero_zero_int)))) # label(fact_1173_split__mult__neg__le) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2154 (all M_26 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat)),M_26) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,M_26),M_26)) # label(fact_412_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2155 (all N_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(suc,N_1)))) # label(fact_2819_lessI) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2156 (all N hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_eq_nat),N))) = hAPP_nat_nat(suc,N)) # label(fact_4744_card__Collect__le__nat) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2157 (all C all X_2 all Ta all A all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),D)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),D))),Ta))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Ta)))))) # label(fact_668_zdvd__period) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2158 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) -> zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N_1))) # label(fact_2721_div__less) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2159 (all K_2 hAPP_int_rat(hAPP_int_fun_int_rat(fract,K_2),one_one_int) = hAPP_int_rat(ring_1_of_int_rat,K_2)) # label(fact_4677_of__int__rat) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2160 (all A_3 all B_2 hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) = hAPP_P1175774780nt_int(product_fst_int_int,divmod_int(A_3,B_2))) # label(fact_4223_div__int__def) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2161 (all B_1_1 all B_2_1 all B_3_1 is_bool(tendsto_nat_complex(B_1_1,B_2_1,B_3_1))) # label(gsy_c_Limits_Otendsto_000tc__Nat__Onat_000tc__Complex__Ocomplex) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2162 (all A_109 all B_88 all C_48 (is_int(C_48) & is_int(B_88) -> (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_109),C_48) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_109),B_88) -> C_48 = B_88))) # label(fact_1925_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2163 (all B_126 all A_150 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_150)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_126),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_150),B_126)),zero_zero_real))))) # label(fact_1235_mult__nonneg__nonpos) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2164 (all B_96 all A_117 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_117),zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_96),zero_zero_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_117),B_96)),zero_zero_nat))))) # label(fact_1864_add__neg__nonpos) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2165 (all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_2769_div__2__gt__zero) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2166 (all X_1 all Y_1 hAPP_nat_nat(nat_gcd(X_1),Y_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Y_1)) # label(fact_4611_gcd__eq__nitpick__gcd) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2167 (all A_187 all N_37 all N_36 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_37),N_36)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_187)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_187),N_37)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_187),N_36)))))) # label(fact_892_power__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2168 (all Lx_4 all Ly_2 all Rx_4 all Ry_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_4),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ly_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_4),Ry_2))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_4),Ly_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_4),Ry_2))) # label(fact_152_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2169 (all B_1_1 all B_2_1 is_bool(hAPP_P603027463t_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2170 (all N all S (hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,S)) -> ((all X (hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X),S)) -> hBOOL(hAPP_f448129468l_bool(finite_finite_int,X)))) -> ((all X (hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X),S)) -> N = hAPP_f957591787ol_nat(finite_card_int,X))) -> hAPP_f521865025ol_nat(hAPP_f93439879ol_nat(big_co1758102091ol_nat,cOMBK_1643584600t_bool(N)),S) = hAPP_f521865025ol_nat(hAPP_f93439879ol_nat(big_co1758102091ol_nat,finite_card_int),S))))) # label(fact_5027_card__setsum__aux) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2171 (all N all Q_3 all P_1 ((all X hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(Q_3,X)),hAPP_nat_nat(P_1,X)))) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,P_1),hAPP_n1699378549t_bool(ord_lessThan_nat,N))),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,Q_3),hAPP_n1699378549t_bool(ord_lessThan_nat,N))) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f1914919701at_nat(hAPP_f1408247010at_nat(cOMBS_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,minus_minus_nat),P_1)),Q_3)),hAPP_n1699378549t_bool(ord_lessThan_nat,N)))) # label(fact_4955_sum__diff__distrib) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2172 (all A_22 all C_7 all B_19 (B_19 != zero_zero_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_7),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_22),B_19)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_22),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_19),C_7))),B_19))) # label(fact_2644_div__mult__self2) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2173 (all C_44 all D_8 all A_101 all B_84 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_101),B_84)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_44),D_8)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_101),C_44)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_84),D_8)))))) # label(fact_2011_add__mono) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2174 (all A_228 A_228 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_228),one_one_real)) # label(fact_329_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2175 (all A_41 all A_40 all B_33 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_41),A_40)),B_33) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_41),B_33)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_40),B_33))) # label(fact_2483_mult_Oadd__left) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2176 (all A_3 all B_2 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),B_2)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),A_3)),B_2))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_1622_zdiff__power2) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2177 (all X_40 all Y_32 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_40),X_40)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_32),Y_32))))) # label(fact_728_sum__squares__ge__zero) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2178 (all M all K_2 all N_1 (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,K_2),N_1) -> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N_1) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),N_1))) # label(fact_5001_gcd__mult__cancel__int) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2179 (all C_44 all D_8 all A_101 all B_84 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_101),B_84)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_44),D_8)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_101),C_44)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_84),D_8)))))) # label(fact_2009_add__mono) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2180 (all X_2 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma)) -> (hAPP_int_int(legacy_zgcd(X_2),Ma) = one_one_int -> hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(noXRRset(Ma),X_2)) = hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(norRRset,Ma))))) # label(fact_4586_card__nor__eq__noX) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2181 (all B_1 all A_1 (hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1)) -> (hBOOL(hAPP_f448129468l_bool(nat_nat_set,B_1)) -> (A_1 = B_1 <-> image_int_nat(nat,A_1) = image_int_nat(nat,B_1))))) # label(fact_4615_transfer__int__nat__set__relations_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2182 (all C_64 all D_11 all A_130 all B_109 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_130),B_109)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_64),D_11)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_130)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_64)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_130),C_64)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_109),D_11)))))))) # label(fact_1419_mult__less__le__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2183 (all K_2 hAPP_int_int(pred,K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),min)) # label(fact_4046_add__Min__right) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2184 (all X_1 of_real_complex(X_1) = hAPP_complex_complex(cnj,of_real_complex(X_1))) # label(fact_4079_complex__cnj__complex__of__real) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2185 (all A (hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially)) -> ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(A,N_2)))) -> ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A,hAPP_nat_nat(suc,N_2))),hAPP_nat_real(A,N_2)))) -> hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A))))))) # label(fact_5064_summable) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2186 (all A all B all Ma (one_one_int = hAPP_int_int(legacy_zgcd(B),Ma) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(bnorRset(A),Ma))))))) # label(fact_4527_Bnor__mem__if) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2187 (all K_2 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),min) = hAPP_int_int(succ,K_2)) # label(fact_3300_diff__bin__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2188 (all A_231 all N_68 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_231),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_68)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_231),N_68)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_281_power__even__eq) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2189 (all F all K_3 all N ((all P_4 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,P_4),N)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,P_4)),K_3)))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),K_3))))) # label(fact_4941_real__setsum__nat__ivl__bounded) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2190 (all A_3 all B_2 all Q_2 all R_3 (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_2)),R_3) = A_3 -> ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_3)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),B_2))) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_3),zero_zero_int)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),R_3))) -> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3)))))) # label(fact_2803_divmod__int__relI) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2191 (all A all N (hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A),N) = zero_zero_rat <-> A = zero_zero_rat & zero_zero_nat != N)) # label(fact_552_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2192 (all F all G ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_nat_real(F,N_2))),hAPP_nat_real(G,N_2)))) -> (hBOOL(summable_real(G)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f352196356l_real(suminf_real,F)),hAPP_f352196356l_real(suminf_real,G))) & hBOOL(summable_real(F))))) # label(fact_4856_summable__le2) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2193 (all M all N_1 hAPP_P1654798560at_nat(product_snd_nat_nat,divmod_nat(M,N_1)) = hAPP_nat_nat(div_mod_nat(M),N_1)) # label(fact_4239_mod__nat__def) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2194 (all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),Y_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arccos,Y_1)),pi)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(arccos,Y_1)))))) # label(fact_3617_arccos__lt__bounded) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2195 (all Ma all K_1 all N (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,K_1),N)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,K_1),hAPP_C498520661umeral(div_mo1740067990umeral(Ma),N))) <-> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,K_1),Ma))))) # label(fact_2390_dvd__mod__iff) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2196 (all C_73 all A_154 all B_130 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_154),B_130)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_73)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_154),C_73)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_130),C_73)))))) # label(fact_1212_mult__right__mono) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2197 (all A_177 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_177),zero_z126310315umeral) = zero_z126310315umeral) # label(fact_1039_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2198 (all M_22 all A_214 all N_62 all B_159 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_214),N_62)),B_159)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_22),N_62)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_214),M_22)),B_159))))) # label(fact_568_power__le__dvd) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2199 (all X_34 all Y_26 -hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_34),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_26),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),zero_zero_rat))) # label(fact_853_not__sum__power2__lt__zero) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2200 (all M hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),M)))) # label(fact_1787_le__square) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2201 (all X_1 all A_3 hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),hAPP_real_real(uminus_uminus_real,A_3)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3))) # label(fact_4435_powr__minus__divide) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2202 (all C_43 all A_100 all B_83 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_100),B_83)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_43),A_100)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_43),B_83))))) # label(fact_2015_add__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2203 (all C_78 all D_14 all A_159 all B_135 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_159),B_135)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_78),D_14)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_159)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_78)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_159),C_78)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_135),D_14)))))))) # label(fact_1188_mult__mono_H) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2204 (all A_3 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),B_2)) -> hAPP_P1975530577nt_int(adjust(B_2),posDivAlg(A_3,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_2))) = posDivAlg(A_3,B_2)) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),B_2)) -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),A_3) = posDivAlg(A_3,B_2)))) # label(fact_2502_posDivAlg__eqn) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2205 (all N_1 (hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1)) -> (exists X hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X)))) # label(fact_3796_even__square) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2206 (all R_3 exists N_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N_2),one_one_int)))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N_2)),R_3)) & is_int(N_2))) # label(fact_4478_real__lb__ub__int) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2207 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> (zero_zero_real = hAPP_real_real(sqrt,X_1) -> X_1 = zero_zero_real))) # label(fact_3322_real__sqrt__eq__zero__cancel) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2208 (all Y_1 all X_1 (hBOOL(hAPP_int_bool(nat_is_nat,X_1)) -> (hBOOL(hAPP_int_bool(nat_is_nat,Y_1)) -> hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_1)))))) # label(fact_5133_Nat__Transfer_Otransfer__int__nat__function__closures_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2209 (all I all J_2 hAPP_int_int(legacy_zgcd(J_2),I) = hAPP_int_int(legacy_zgcd(I),J_2)) # label(fact_4492_zgcd__commute) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2210 (all A_112 all B_91 all C_51 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_112),B_91)),C_51) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_112),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_91),C_51))) # label(fact_1906_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2211 (all X_2 hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),X_2))) = X_2) # label(fact_4764_card__bdd__nat__set__l) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2212 (all Ma all N (is_int(Ma) & is_int(N) -> (hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ma),N) = one_one_int <-> N = one_one_int & Ma = one_one_int | hAPP_int_int(number_number_of_int,min) = Ma & N = hAPP_int_int(number_number_of_int,min)))) # label(fact_968_zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2213 (all A_12 all B_9 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_12),B_9)),B_9)),hAPP_nat_nat(div_mod_nat(A_12),B_9)) = A_12) # label(fact_2709_mod__div__decomp) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2214 (all B_138 all A_162 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_138),zero_z891286103de_int)) & hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_162)) | hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_162),zero_z891286103de_int)) & hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_138)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_162),B_138)),zero_z891286103de_int)))) # label(fact_1177_split__mult__neg__le) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2215 (all B all A all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int))) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(wset(A),P_5))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B)))))) # label(fact_2239_wset__g__1) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2216 (all X_1 all Y_1 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_complex_real(re,X_1)),hAPP_complex_real(re,Y_1)) = hAPP_complex_real(re,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_1),Y_1))) # label(fact_4125_Re_Odiff) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2217 (all N_1 all X_1 (-hBOOL(hAPP_int_bool(even_odd_even_int,X_1)) -> -hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),N_1))))) # label(fact_2895_odd__pow) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2218 (all B_55 all A_63 hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_55),A_63)),B_55) = hAPP_Q1762011733de_int(div_mo231679042de_int(A_63),B_55)) # label(fact_2349_mod__add__self1) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2219 (all N_1 exists P_4 (hBOOL(hAPP_nat_bool(prime,P_4)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),P_4)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_4),hAPP_nat_nat(suc,hAPP_nat_nat(fact,N_1)))))) # label(fact_4569_euclid__bound) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2220 (all A_176 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,zero_z891286103de_int),A_176) = zero_z891286103de_int) # label(fact_1048_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2221 (all A_165 all B_140 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_165),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_165),B_140)))) # label(fact_1138_dvd__triv__left) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2222 (all X_2 hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_eq_nat),X_2))))) # label(fact_4743_bdd__nat__set__le__finite) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2223 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(sin,X_1)))))) # label(fact_3725_sin__gt__zero2) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2224 (all Y_2 all X_2 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Y_2)),zEven))))) # label(fact_2899_EvenOdd_Oeven__plus__even) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2225 (all A_179 all B_151 (hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_179),B_151) = zero_zero_real -> zero_zero_real = A_179 | zero_zero_real = B_151)) # label(fact_1020_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2226 (all A_184 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_184),zero_zero_real))) # label(fact_911_dvd__0__right) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2227 (all K_2 hAPP_int_int(bit1,K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),K_2)),K_2)) # label(fact_447_Bit1__def) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2228 (all X_1 all N_1 hAPP_int_real(real_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),N_1)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(real_int,X_1)),N_1)) # label(fact_4390_real__of__int__power) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2229 (all X_1 all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1)) -> (exists R_1 exists A_2 (hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(sin,A_2)) = Y_1 & X_1 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(cos,A_2)))))) # label(fact_3947_polar__ex1) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2230 (all A all B all C (C != zero_zero_real -> (hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),A) <-> B = A))) # label(fact_1599_real__mult__left__cancel) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2231 (all N (N = zero_zero_nat <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),zero_zero_nat)))) # label(fact_1751_le__0__eq) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2232 (all B_2 all A_3 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(B_2),zero_zero_int),P_3)) & -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),P_3)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)),zero_zero_int),P_3)))))) # label(fact_1500_zcong__zprime__prod__zero__contra) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2233 (all S ((all M_1 exists N_2 (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,N_2),S)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2)))) <-> -hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S)))) # label(fact_4608_infinite__nat__iff__unbounded) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2234 (all N_13 all A_7 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_7)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_7),hAPP_nat_nat(suc,N_13)))))) # label(fact_3122_power__gt1) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2235 (all K_2 all L_2 hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),L_2)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit1,K_2)),hAPP_int_int(bit0,L_2))) # label(fact_445_add__Bit1__Bit0) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2236 (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2)) -> hBOOL(hAPP_real_bool(deriv_real(ln,X_2),hAPP_real_real(inverse_inverse_real,X_2))))) # label(fact_3865_DERIV__ln) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2237 (all Ta all D all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),D_1)) -> (all X all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Ta))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D_1))),Ta))))))) # label(fact_1816_inf__period_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2238 (all A_133 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_133),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_133),one_one_rat)))) # label(fact_1373_less__add__one) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2239 (all Lx_2 all Ly all Rx_2 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_2),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Ly),Rx_2)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_2),Ly)),Rx_2)) # label(fact_169_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2240 (all A_47 zero_z891286103de_int = hAPP_Q1762011733de_int(div_mo231679042de_int(A_47),one_on1684967323de_int)) # label(fact_2418_mod__by__1) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2241 (all A_237 all B_166 all C_97 all D_20 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_237),C_97)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_166),D_20)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_237),B_166)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_97),D_20))) # label(fact_198_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2242 (all X_1 all Y_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,X_1),Y_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_1),Y_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_4007_coprime__sos) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2243 (all B all A all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int))) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(wset(A),P_5))) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(inv(P_5),B)),hAPP_i1948725293t_bool(wset(A),P_5)))))))) # label(fact_2229_wset__mem__inv__mem) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2244 -(all S1 (is_int(S1) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,S1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))))) # label(fact_918__096_B_Bthesis_O_A_I_B_Bs1_O_A_091s1_A_094_A2_A_061_A_N1_093_A_Imod_A4_) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2245 (all Y_2 all X_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),Y_2)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(nat,X_2)),hAPP_int_nat(nat,Y_2))))))) # label(fact_2742_transfer__nat__int__relations_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2246 (all A all B all C all D (hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,C),D) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C),D)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A),B))))) # label(fact_2099_diff__eq__diff__less) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2247 (all B_40 all A_48 hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_40),A_48)),B_40) = zero_z126310315umeral) # label(fact_2413_mod__mult__self1__is__0) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2248 (all K_2 hAPP_int_int(bit0,hAPP_int_int(uminus_uminus_int,K_2)) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(bit0,K_2))) # label(fact_3512_minus__Bit0) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2249 (all K_1 all L (is_int(L) & is_int(K_1) -> (L = K_1 <-> hAPP_int_int(bit1,L) = hAPP_int_int(bit1,K_1)))) # label(fact_258_rel__simps_I51_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2250 (all P all Q all R hAPP_r373117745t_bool(hAPP_f1115375056t_bool(hAPP_f271976045t_bool(cOMBC_1801450329t_bool,P),Q),R) = hAPP_f289318786t_bool(hAPP_r653725362t_bool(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__O_021) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2251 (all A_3 all B_2 hAPP_complex_complex(uminus473333897omplex,hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),B_2)) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(uminus_uminus_real,A_3)),hAPP_real_real(uminus_uminus_real,B_2))) # label(fact_3962_complex__minus) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2252 (all A_106 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_106),zero_z891286103de_int) = A_106) # label(fact_1975_add__0__right) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2253 (all Z_1 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Z_1),hAPP_complex_complex(cnj,Z_1)) = of_real_complex(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,Z_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(im,Z_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_4110_complex__mult__cnj) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2254 (all A_209 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_209),zero_zero_nat) = zero_zero_nat) # label(fact_616_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2255 (all B all A all C (hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C),A) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B),A) <-> B = C)) # label(fact_1931_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2256 (all N_1 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,M),N_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(fact_fact_int,M)),hAPP_int_int(fact_fact_int,N_1)))))) # label(fact_4273_fact__less__mono__int) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2257 (all N_41 all A_190 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),A_190)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_190),N_41))))) # label(fact_860_one__le__power) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2258 (all M_8 all N_22 hAPP_n1108039445umeral(semiri1619134803umeral,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_8),N_22)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_n1108039445umeral(semiri1619134803umeral,M_8)),hAPP_n1108039445umeral(semiri1619134803umeral,N_22))) # label(fact_2955_of__nat__add) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2259 (all B_2 all A_3 (is_int(B_2) & is_int(A_3) -> (A_3 != zero_zero_int | B_2 != zero_zero_int -> one_one_int = hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(legacy_zgcd(A_3),B_2))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),hAPP_int_int(legacy_zgcd(A_3),B_2)))))) # label(fact_4512_div__zgcd__relprime) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2260 (all N_1 N_1 != hAPP_nat_nat(suc,N_1)) # label(fact_2838_Suc__n__not__n) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2261 (all A_72 all B_64 all C_32 hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_72),hAPP_C498520661umeral(div_mo1740067990umeral(B_64),C_32))),C_32) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_72),B_64)),C_32)) # label(fact_2316_mod__mult__right__eq) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2262 (all D all Sa exists Z all X (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X),Z)) -> (-hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,D),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X),Sa))) <-> -hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,D),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X),Sa)))))) # label(fact_2265_minf_I10_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2263 (all P all Q all R hAPP_f472159229t_bool(P,hAPP_r1962736578t_bool(Q,R)) = hAPP_r1623423338t_bool(hAPP_f704188951t_bool(hAPP_f1456711209t_bool(cOMBB_690122963l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__fun_Itc__Int__Oint_) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2264 (all X_1 hAPP_real_real(cos,X_1) = hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X_1))) # label(fact_3732_cos__sin__eq) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2265 (all A_128 all C_62 all B_107 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_128),C_62)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_107),C_62))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_62)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_128),B_107))))) # label(fact_1434_mult__right__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2266 (all A_232 all C_92 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_232),C_92) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_92),A_232)) # label(fact_240_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2267 (all A_1 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma)) -> (hBOOL(hAPP_int_bool(is_RRset(A_1),Ma)) -> hAPP_i1948725293t_bool(norRRset,Ma) = image_int_int(rRset2norRR(A_1,Ma),A_1)))) # label(fact_4596_RRset2norRR__eq__norR) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2268 (all N_1 hAPP_nat_real(real_nat,N_1) = hAPP_int_real(real_int,hAPP_nat_int(semiri1621563631at_int,N_1))) # label(fact_4391_real__of__int__of__nat__eq) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2269 (all A_26 all C_9 all B_23 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,C_9),B_23)) -> hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_26),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,B_23),C_9)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_26),B_23)),C_9))) # label(fact_2593_div__mult__swap) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2270 (exists K_4 ((all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(cnj,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X)),K_4)))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),K_4)))) # label(fact_4086_cnj_Ononneg__bounded) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2271 (all N_1 all V_2 ((-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> hAPP_int_nat(nat,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,V_2)),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(number_number_of_nat,V_2)),N_1)) & (hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(number_number_of_nat,V_2)),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N_1)))) # label(fact_3287_power__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2272 (all K_1 all L (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,K_1)),hAPP_int_int(number_number_of_int,L))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L)))) # label(fact_118_less__number__of__int__code) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2273 (all N_1 all X_1 all Y_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_1),N_1))))) # label(fact_1514_divides__exp) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2274 (all P all Q all R hAPP_f1325804733t_bool(P,hAPP_P2053772734t_bool(Q,R)) = hAPP_P1400155668t_bool(hAPP_f223623489t_bool(hAPP_f1815301987t_bool(cOMBB_650041899nt_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__f) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2275 (all A all B all C (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),A)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B)))))) # label(fact_1404_mult__le__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2276 (all X_2 hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),sin)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_2),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_2)),hAPP_real_real(sin,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_2)),hAPP_real_real(sin,X_2)))))) # label(fact_4794_DERIV__sin__realpow2) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2277 (all A_235 all B_164 all C_95 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_235),B_164)),C_95) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_235),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_164),C_95))) # label(fact_221_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2278 (all M all R_3 all S_1 all T_1 all N_1 (is_int(R_3) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_1)) -> (hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,R_3),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,S_1),T_1)) = xzgcd(M,N_1) -> R_3 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,S_1),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,T_1),N_1)))))) # label(fact_4198_xzgcd__linear) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2279 (all M all N_1 hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,M)),hAPP_nat_real(real_nat,N_1))) # label(fact_3651_real__of__nat__add) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2280 (all R_3 all S_1 all T_1 all R_5 all S_3 all M all T_2 all N_1 (R_5 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,S_3),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,T_2),N_1)) -> (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,S_1),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,T_1),N_1)) = R_3 -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,S_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,R_5),R_3)),S_1))),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,T_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,R_5),R_3)),T_1))),N_1)) = hAPP_int_int(div_mod_int(R_5),R_3)))) # label(fact_2690_xzgcda__linear__aux2) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2281 (all C_29 all A_69 all B_61 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_29),hAPP_nat_nat(div_mod_nat(A_69),B_61)) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_29),A_69)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_29),B_61))) # label(fact_2327_mod__mult__mult1) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2282 (all X_1 all Y_1 (is_int(X_1) -> hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,fTrue),X_1),Y_1) = X_1)) # label(help_If_1_1_If_000tc__Int__Oint_T) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2283 (all M_6 all N_20 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_nat_complex(semiri2020571505omplex,M_6)),N_20) = hAPP_nat_complex(semiri2020571505omplex,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_6),N_20))) # label(fact_2982_of__nat__power) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2284 (all A_61 all B_53 all C_24 hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_61),C_24)),B_53)),C_24) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_61),B_53)),C_24)) # label(fact_2356_mod__add__left__eq) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2285 (all N_54 all A_204 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_204)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_204),N_54))))) # label(fact_664_zero__less__power) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2286 (all A_230 all M_29 all B_163 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_230),B_163)),M_29) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_230),M_29)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_163),M_29))) # label(fact_301_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2287 (all A_173 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,zero_zero_rat),A_173)) -> zero_zero_rat = A_173)) # label(fact_1082_dvd__0__left) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2288 (all I all J_2 all K_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),J_2)),K_2)) # label(fact_1646_diff__diff__left) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2289 (all A_201 all N_50 all N_49 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_50),N_49)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_201)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_201),one_one_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_201),N_49)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_201),N_50))))))) # label(fact_682_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2290 (all X_3 (hBOOL(cauchy_complex(X_3)) -> hBOOL(cauchy_complex(hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,cnj),X_3))))) # label(fact_4755_cnj_OCauchy) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2291 (all X_1 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M)) -> (one_one_int = hAPP_int_int(legacy_zgcd(X_1),M) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),hAPP_int_nat(phi,M))),one_one_int),M))))) # label(fact_4481_Euler__Fermat) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2292 (all Y_1 all X_1 (hBOOL(hAPP_int_bool(even_odd_even_int,X_1)) -> (-hBOOL(hAPP_int_bool(even_odd_even_int,Y_1)) -> -hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Y_1)))))) # label(fact_2891_Parity_Oeven__plus__odd) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2293 (all X_1 all Y_1 hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_1) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(abs_abs_int,X_1)),hAPP_int_int(abs_abs_int,Y_1))) # label(fact_4989_gcd__abs__int) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2294 (all A_181 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_181),A_181))) # label(fact_983_dvd__refl) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2295 (all X_1 all A_3 all B_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),B_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),B_2)))))) # label(fact_4409_powr__less__mono) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2296 (all A_239 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_239),A_239) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_239),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_42_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2297 (all P all Q all R hAPP_f54304608l_bool(P,hAPP_i418383825t_bool(Q,R)) = hAPP_int_bool(hAPP_f41603166t_bool(hAPP_f389116883t_bool(cOMBB_1593174431ol_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obo) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2298 (all Ma all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)) <-> (exists K N = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),K)))) # label(fact_1781_le__iff__add) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2299 (all X_1 all Y_1 all Z_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Z_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y_1),Z_1))) # label(fact_4309_gcd__semilattice__nat_Oinf__left__commute) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2300 (all M_9 all N_23 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,M_9)),hAPP_nat_real(semiri132038758t_real,N_23))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_23)))) # label(fact_2946_of__nat__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2301 (all B_1_1 all B_2_1 all B_3_1 is_bool(tendst1507391555omplex(B_1_1,B_2_1,B_3_1))) # label(gsy_c_Limits_Otendsto_000tc__Complex__Ocomplex_000tc__Complex__Ocomplex) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2302 (all A_104 A_104 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A_104)) # label(fact_1989_add__0__left) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2303 (all A_220 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_220),hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit1,pls))) = A_220) # label(fact_459_mult__numeral__1__right) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2304 (all Ma all N (one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),N) <-> N = one_one_nat & one_one_nat = Ma)) # label(fact_1796_nat__mult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2305 (all M zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),zero_zero_nat)) # label(fact_1754_mult__0__right) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2306 (all B_69 all A_80 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_80)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_69)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_80),B_69)))))) # label(fact_2194_add__pos__pos) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2307 (all A_194 all B_153 all N_46 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_194),N_46)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,B_153),N_46)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_194),B_153)),N_46)) # label(fact_817_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2308 (all A_3 all B_2 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,expi(A_3)),expi(B_2)) = expi(hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_3),B_2))) # label(fact_3999_expi__add) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2309 (all Y_1 exists X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) & hAPP_real_real(tan,X) = Y_1)) # label(fact_3627_lemma__tan__total1) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2310 (all A all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),Ma)) -> hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,hAPP_i1948725293t_bool(bnorRset(A),Ma)),rsetR(Ma))))) # label(fact_4595_Bnor__in__RsetR) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2311 (all X_1 one_one_real = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sin,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3727_sin__cos__squared__add) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2312 (all X_7 all Q_5 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_7),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_7),Q_5)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_7),hAPP_nat_nat(suc,Q_5))) # label(fact_3022_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2313 (all V_20 all V_19 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_19)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_20)) -> hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_19),V_20)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_19)),hAPP_int_real(number267125858f_real,V_20))))) # label(fact_374_semiring__add__number__of) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2314 (all N_1 hAPP_nat_nat(suc,N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),one_one_nat)) # label(fact_3072_Suc__eq__plus1) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2315 (all Z_1 all W (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z_1)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),W)))) # label(fact_64_zle__linear) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2316 (all A_237 all B_166 all C_97 all D_20 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_237),B_166)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_97),D_20)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_237),C_97)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_166),D_20))) # label(fact_193_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2317 (all P all Q all R hAPP_int_int(hAPP_P2110489235nt_int(P,R),hAPP_P1175774780nt_int(Q,R)) = hAPP_P1175774780nt_int(hAPP_f521271395nt_int(hAPP_f1914742907nt_int(cOMBS_2019229620nt_int,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oin) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2318 (all G all F all X_2 all D (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D)) -> ((all Z (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_2))),D)) -> hAPP_real_real(G,hAPP_real_real(F,Z)) = Z)) -> ((all Z (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_2))),D)) -> hBOOL(hAPP_real_bool(isCont_real_real(F),Z)))) -> (exists Z (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_2))),D)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,Z)),hAPP_real_real(F,X_2))))))))) # label(fact_5180_lemma__isCont__inj2) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2319 (all Z_1 exists R_1 exists A_2 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_1)),hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(cos,A_2)),hAPP_real_real(sin,A_2))) = Z_1) # label(fact_4001_complex__split__polar) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2320 (all X_1 all A_3 all B_2 (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_1),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_1),B_2)) -> one_one_int = hAPP_int_int(abs_abs_int,X_1))))) # label(fact_5015_coprime__common__divisor__int) # label(axiom) # label(non_clause). [assumption]. 5.56/5.63 2321 (all A_187 all N_37 all N_36 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_37),N_36)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_187)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_187),N_37)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_187),N_36)))))) # label(fact_894_power__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2322 (all M_13 hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,M_13)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,M_13))) # label(fact_2840_of__nat__Suc) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2323 (all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N)) <-> hBOOL(hAPP_nat_bool(even_odd_even_nat,N)))) # label(fact_3783_even__dvd) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2324 (all B_74 all A_85 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_85),zero_zero_rat)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_74),zero_zero_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_85),B_74)),zero_zero_rat))))) # label(fact_2153_add__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2325 (all K_1 hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_lessThan_nat,K_1)))) # label(fact_4948_finite__lessThan) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2326 (all Z_1 Z_1 = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_complex_real(re,Z_1)),hAPP_complex_real(im,Z_1))) # label(fact_4119_complex__surj) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2327 (all X_2 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,X_2)),hAPP_real_real(exp_real,Y_2))))) # label(fact_3822_exp__le__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2328 (all P all Q all R hAPP_f151095827nt_int(P,hAPP_i1216375074nt_int(Q,R)) = hAPP_i1836591008nt_int(hAPP_f400200275nt_int(hAPP_f1731308757nt_int(cOMBB_735549356nt_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__p) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2329 (all A_86 all B_75 A_86 = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_86),B_75)),B_75)) # label(fact_2138_diff__add__cancel) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2330 (all P_1 all I_1 all J (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J)) -> (hBOOL(hAPP_nat_bool(P_1,J)) -> ((all I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J)) -> (hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,I_2))) -> hBOOL(hAPP_nat_bool(P_1,I_2))))) -> hBOOL(hAPP_nat_bool(P_1,I_1)))))) # label(fact_3260_inc__induct) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2331 (all X_2 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_nat_bool(even_odd_even_nat,N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),zero_zero_real)) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(root,N),X_2),hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_nat_real(real_nat,N))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,zero_zero_nat))))))))))) # label(fact_3893_DERIV__even__real__root) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2332 (all Y_21 all X_27 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_27)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_21)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_21),one_one_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_27),Y_21)),X_27)))))) # label(fact_1470_mult__right__le__one__le) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2333 (all P_1 all Ma all N (hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,Ma),N))) <-> (exists Q_4 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),Q_4)),Ma)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),hAPP_nat_nat(suc,Q_4)))) & hBOOL(hAPP_nat_bool(P_1,Q_4)))) | N = zero_zero_nat & hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)))) # label(fact_3212_split__div_H) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2334 (all K_2 hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,one_one_rat),hAPP_int_rat(number_number_of_rat,K_2)) = frct(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,one_one_int),hAPP_int_int(number_number_of_int,K_2)))) # label(fact_4645_Frct__code__post_I5_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2335 (all A_110 all B_89 all C_49 (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_110),B_89) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_110),C_49) -> C_49 = B_89)) # label(fact_1920_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2336 (all N (is_int(N) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N)) -> hAPP_nat_int(semiri1621563631at_int,hAPP_f957591787ol_nat(finite_card_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_eq_int),N))))) = N))) # label(fact_4734_int__card__bdd__int__set__l__le) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2337 (all Ta all B_1 all D all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),D_1)) -> (all X (is_int(X) -> ((all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B_1)) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) != X)))))) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Ta))) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),D_1)),Ta))))))))) # label(fact_5102_bset_I10_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2338 (all V_23 hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_23),hAPP_int_int(bit1,pls))) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_23)),one_one_complex)) # label(fact_56_add__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2339 (all A_171 all B_146 all C_84 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_171),B_146)),C_84)) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_171),C_84)))) # label(fact_1094_dvd__mult__left) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2340 (all N all Ma (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,N),Ma)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,quickcheck_int_of(N)),quickcheck_int_of(Ma))))) # label(fact_4252_less__code__int__def) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2341 (all X_2 all Y_2 (Y_2 = zero_zero_real & X_2 = zero_zero_real <-> zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_2),X_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_2),Y_2)))) # label(fact_1617_real__two__squares__add__zero__iff) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2342 (all A (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int)))) # label(fact_2178_double__add__le__zero__iff__single__add__le__zero) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2343 (all A_203 all N_53 all B_155 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_203),N_53)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_155),N_53))) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_155)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_203),B_155))))) # label(fact_674_power__less__imp__less__base) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2344 (all B_1_1 all B_2_1 is_bool(hAPP_f54304608l_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2345 (all A_102 all C_45 all B_85 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_102),C_45)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_85),C_45))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_102),B_85)))) # label(fact_2002_add__le__imp__le__right) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2346 (all N_1 all D_3 all A_3 (one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),A_3) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_3),N_1)) = one_one_nat)) # label(fact_4361_coprime__exp__nat) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2347 (all C_8 all A_24 all B_21 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_24),B_21)) -> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,B_21),A_24)),C_8) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_21),C_8)),A_24))) # label(fact_2602_dvd__div__mult) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2348 (all A_3 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_3)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3)))))) # label(fact_4431_ge__one__powr__ge__zero) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2349 (all P_1 all N all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) -> hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N))),hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B),A)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N)))) # label(fact_4241_Bolzano__bisect__diff) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2350 (all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_3)) -> (exists P_4 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_4),A_3)) & hBOOL(hAPP_int_bool(zprime,P_4)) & is_int(P_4))))) # label(fact_1836_zprime__factor__exists) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2351 (all B_2 all B_5 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_5),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_5)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2))))))) # label(fact_2625_zdiv__mono2__neg) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2352 (all X_2 (hAPP_int_int(div_mod_int(X_2),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = one_one_int <-> hAPP_int_int(div_mod_int(X_2),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) != zero_zero_int)) # label(fact_2468_neq__one__mod__two) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2353 (all A_16 all B_13 A_16 = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_16),B_13)),B_13)),hAPP_C498520661umeral(div_mo1740067990umeral(A_16),B_13))) # label(fact_2677_mod__div__equality) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2354 (all N_26 hAPP_nat_rat(semiri151668891at_rat,N_26) = hAPP_int_rat(number_number_of_rat,hAPP_nat_int(semiri1621563631at_int,N_26))) # label(fact_2828_number__of__int) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2355 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) -> (M != N_1 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))))) # label(fact_1768_le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2356 (all A_148 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_148),A_148)))) # label(fact_1253_zero__le__square) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2357 (all A_14 all B_11 all C_4 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_14),B_11)),B_11)),hAPP_int_int(div_mod_int(A_14),B_11))),C_4) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_14),C_4)) # label(fact_2683_div__mod__equality) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2358 (all B_74 all A_85 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_85),zero_z891286103de_int)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_74),zero_z891286103de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_85),B_74)),zero_z891286103de_int))))) # label(fact_2152_add__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2359 (all Q_2 -(all A_2 all B_4 (is_int(A_2) & is_int(B_4) -> (hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_4) = Q_2 -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_4)) -> one_one_int != hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_2),B_4)))))) # label(fact_5025_Rat__cases) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2360 (all A_16 all B_13 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_16),B_13)),B_13)),hAPP_Q1762011733de_int(div_mo231679042de_int(A_16),B_13)) = A_16) # label(fact_2678_mod__div__equality) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2361 (all N_48 all A_199 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_199)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_199),one_on1684967323de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_199),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_199),N_48))),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_199),N_48)))))) # label(fact_724_power__Suc__less) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2362 (all A_183 one_one_rat = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_183),zero_zero_nat)) # label(fact_961_power__0) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2363 (all Z_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_int_nat(nat,Z_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Z_2)))) # label(fact_3201_one__less__nat__eq) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2364 (all X_44 all Y_36 all Z_15 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,X_44),Y_36)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,X_44),Z_15)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,X_44),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,Y_36),Z_15))) # label(fact_320_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2365 (all A_35 zero_z891286103de_int = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,zero_z891286103de_int),A_35)) # label(fact_2544_div__0) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2366 (all A_3 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> (exists X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) & (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y)) & hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),N_1) = A_3 -> X = Y)) & hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N_1) = A_3))))) # label(fact_1629_realpow__pos__nth__unique) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2367 (all F all G ((exists N_3 all N_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_3),N_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_nat_real(F,N_2))),hAPP_nat_real(G,N_2))))) -> (hBOOL(summable_real(G)) -> hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,abs_abs_real),F)))))) # label(fact_4828_summable__rabs__comparison__test) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2368 (all X_1 all A_3 archim1246769320r_real(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_int_real(real_int,A_3))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,archim1246769320r_real(X_1)),A_3)) # label(fact_4419_floor__add) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2369 (all Ma all N (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(semiri151668891at_rat,Ma)),hAPP_nat_rat(semiri151668891at_rat,N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)))) # label(fact_2952_of__nat__le__iff) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2370 (all X_1 all Y_1 (is_int(X_1) & is_int(Y_1) -> Y_1 = X_1 | -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,X_1),Y_1)))) # label(help_fequal_1_1_fequal_000tc__Int__Oint_T) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2371 (all Y_2 all X_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X_2)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zOdd)) -> (exists N_2 (is_int(N_2) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Y_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N_2),X_2))))),X_2))))))) # label(fact_3474_best__odd__division__abs) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2372 (all B_58 all B_57 all A_66 all C_26 all A_65 (hAPP_Q1762011733de_int(div_mo231679042de_int(A_65),C_26) = hAPP_Q1762011733de_int(div_mo231679042de_int(A_66),C_26) -> (hAPP_Q1762011733de_int(div_mo231679042de_int(B_58),C_26) = hAPP_Q1762011733de_int(div_mo231679042de_int(B_57),C_26) -> hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_66),B_58)),C_26) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_65),B_57)),C_26)))) # label(fact_2341_mod__mult__cong) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2373 (all A_89 A_89 = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,one_one_rat),A_89)) # label(fact_2120_mult__1) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2374 (all N_41 all A_190 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,one_on1684967323de_int),A_190)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,one_on1684967323de_int),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_190),N_41))))) # label(fact_864_one__le__power) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2375 (all B_2 all A_3 (is_int(A_3) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),A_3)) -> hAPP_int_int(div_mod_int(A_3),B_2) = A_3)))) # label(fact_2448_mod__neg__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2376 (all A all B all C (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),A)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B))))) # label(fact_1320_mult__less__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2377 (all N_54 all A_204 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_204)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_204),N_54))))) # label(fact_663_zero__less__power) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2378 (all X_4 all Y_4 hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_4),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_real_real(sin,hAPP_real_real(arccos,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_4),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))))) # label(fact_3745_sin__arccos__lemma1) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2379 (all N_1 all B_2 all A_3 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,P_3),N_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,P_3),N_1)),B_2)))))) # label(fact_726_zprime__power__zdvd__cancel__left) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2380 (all Lx_6 all Ly_4 all Rx_6 all Ry_4 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_6),Ly_4)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Rx_6),Ry_4)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_6),Rx_6)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Ly_4),Ry_4))) # label(fact_136_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2381 (all A_3 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M)) -> (exists X (is_int(X) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),X),M)) & (all Y (is_int(Y) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),Y),M)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),M)) -> Y = X))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),M)))))) # label(fact_1837_zcong__zless__unique) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2382 (all C_80 all A_164 all B_139 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_164),B_139)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_164),C_80)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_164),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_139),C_80)))))) # label(fact_1144_dvd__add) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2383 (all A all B all A_1 all Ma (is_int(A) & is_int(B) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma)) -> (hBOOL(hAPP_int_bool(is_RRset(A_1),Ma)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),A_1)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),A_1)) -> B = A))))))) # label(fact_4589_RRset__zcong__eq) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2384 (all N_60 all X_41 all Y_33 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_41),Y_33)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_41),N_60)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_33),N_60))))) # label(fact_588_dvd__power__same) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2385 (all X_49 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_49),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_49),X_49)) # label(fact_32_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2386 (all X_39 all Y_31 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_39),X_39)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_31),Y_31))),zero_zero_int))) # label(fact_735_not__sum__squares__lt__zero) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2387 (all A_90 A_90 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_90),one_one_nat)) # label(fact_2108_mult__1__right) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2388 (all C_59 all A_125 all B_104 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_59),A_125)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_59),B_104))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_59)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_125),B_104))))) # label(fact_1451_mult__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2389 (all X_2 all Y_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2)) <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_2),Y_2) = X_2)) # label(fact_4328_gcd__semilattice__nat_Ole__iff__inf) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2390 (all X_2 all P_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),hAPP_i1948725293t_bool(sRStar,P_5))) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(standardRes(P_5),hAPP_int_int(multInv(P_5),X_2))),hAPP_i1948725293t_bool(sRStar,P_5))))))) # label(fact_2505_StandardRes__SRStar__prop2) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2391 (all X_1 all Y_1 hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_1),Y_1) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),hAPP_real_real(inverse_inverse_real,Y_1))) # label(fact_3830_divide__real__def) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2392 (all K_2 all I all J_2 (hAPP_int_int(legacy_zgcd(I),J_2) = one_one_int -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),J_2))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),K_2))))) # label(fact_4517_zrelprime__dvd__mult) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2393 (all M_13 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_nat_real(semiri132038758t_real,M_13)) = hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(suc,M_13))) # label(fact_2842_of__nat__Suc) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2394 (all Z_9 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_9),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_9),Z_9)) # label(fact_501_mult__2__right) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2395 (all X_2 all Y_2 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Y_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_2),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,Y_2),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3235_even__div__2__l) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2396 (all X_2 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X_2)),zero_zero_real))))) # label(fact_3919_real__root__lt__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2397 (all A_3 all B_2 all C_1 hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,B_2),C_1)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)),C_1)) # label(fact_4981_gcd__assoc__int) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2398 (all A_75 hAPP_Q1762011733de_int(div_mo231679042de_int(A_75),A_75) = zero_z891286103de_int) # label(fact_2305_mod__self) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2399 (all Ma all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,Ma)),hAPP_nat_real(semiri132038758t_real,N))))) # label(fact_2934_of__nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2400 (all A all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,B),A)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A),B)))) # label(fact_4015_coprime__commute) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2401 (all X_1 all Y_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) -> -(hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))))) # label(fact_1718_dvd_Oless__asym) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2402 (all X_40 all Y_32 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_40),X_40)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_32),Y_32))))) # label(fact_730_sum__squares__ge__zero) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2403 (all Y_1 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1)) -> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(ln,X_1)),hAPP_real_real(ln,Y_1)) = hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),Y_1))))) # label(fact_3453_ln__mult) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2404 (all D (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D),zero_zero_nat)) <-> D = one_one_nat)) # label(fact_4024_coprime__0) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2405 (all A_203 all N_53 all B_155 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_203),N_53)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_155),N_53))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_155)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_203),B_155))))) # label(fact_672_power__less__imp__less__base) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2406 (all I_1 all M_2 (-hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,zero_zero_nat),M_2)) -> hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),M_2))),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),hAPP_nat_nat(suc,I_1))))) = hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,hAPP_f66927821l_bool(hAPP_f2037783381l_bool(cOMBB_1146692694ol_nat,member_nat),suc)),M_2))),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),I_1)))))) # label(fact_4767_card__less__Suc2) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2407 (all X_1 all Y_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) -> X_1 = Y_1))) # label(fact_1703_dvd_Oantisym) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2408 (all N_63 all M_23 all X_42 all Y_34 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,X_42),Y_34)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_63),M_23)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_42),N_63)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_34),M_23)))))) # label(fact_561_dvd__power__le) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2409 (all A_236 all B_165 all C_96 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_236),C_96)),B_165) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_236),B_165)),C_96)) # label(fact_212_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2410 (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(ln,X_2))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X_2))))) # label(fact_3452_ln__ge__zero__iff) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2411 (all C_68 all A_143 all B_120 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_143),B_120)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_68)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_143),C_68)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_120),C_68)))))) # label(fact_1277_mult__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2412 (all N all Ma (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),Ma)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(suc,Ma))) <-> Ma = N))) # label(fact_2860_not__less__less__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2413 (all X_2 hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,fact_fact_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat)))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_2)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat)))),hAPP_real_real(sin,X_2)))) # label(fact_4862_sin__paired) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2414 (all D all Sa exists Z all X (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Z),X)) -> (-hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,D),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X),Sa))) <-> -hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,D),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X),Sa)))))) # label(fact_2259_pinf_I10_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2415 (all V all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))),zero_zero_nat),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))),N)))) # label(fact_5119_min__Suc__number__of) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2416 (all P all Q all R hAPP_real_real(hAPP_r1250527377l_real(P,R),hAPP_real_real(Q,R)) = hAPP_real_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__RealDef_) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2417 (all A all C all B (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A),C)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B),C))) <-> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A),B)))) # label(fact_2072_add__less__cancel__right) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2418 (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,X_2)),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),zero_zero_real)))) # label(fact_3328_real__sqrt__lt__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2419 (all X_2 all Y_2 all B (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),B)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),Y_2)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B),X_2)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B),Y_2)))))) # label(fact_883_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2420 (all B_40 all A_48 zero_z891286103de_int = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_40),A_48)),B_40)) # label(fact_2414_mod__mult__self1__is__0) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2421 (all W_8 hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit1,W_8)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,one_one_rat),hAPP_int_rat(number_number_of_rat,W_8))),hAPP_int_rat(number_number_of_rat,W_8))) # label(fact_451_number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2422 (all X_1 bezw(X_1,zero_zero_nat) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,one_one_int),zero_zero_int)) # label(fact_4154_bezw__0) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2423 (all X_1 all Y_1 all Z_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_1),X_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_1),Y_1)))))) # label(fact_1604_real__mult__less__mono2) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2424 (all X_1 X_1 = hAPP_i1732201573de_int(quickcheck_of_int,quickcheck_int_of(X_1))) # label(fact_4554_int__of__inverse) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2425 (all P all Q all R hAPP_nat_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,P),Q),R) = hAPP_real_real(hAPP_n546711566l_real(P,R),hAPP_nat_real(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__RealDef__Orea) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2426 (all B_125 all A_149 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_149)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_125)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_149),B_125)))))) # label(fact_1241_mult__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2427 (all A_3 all B_2 (is_int(B_2) -> (B_2 != zero_zero_int -> (zero_zero_int = hAPP_int_int(div_mod_int(A_3),B_2) -> hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(uminus_uminus_int,B_2))) & (hAPP_int_int(div_mod_int(A_3),B_2) != zero_zero_int -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2))),one_one_int) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(uminus_uminus_int,B_2)))))) # label(fact_3552_zdiv__zminus2__eq__if) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2428 (all F all A_1 hBOOL(hAPP_f54304608l_bool(nat_tr160667106at_int(F),A_1))) # label(fact_2788_transfer__morphism__def) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2429 (all B_1_1 all B_2_1 is_int(hAPP_nat_int(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__Nat__Onat_000tc__Int__Oint) # label(hypothesis) # label(non_clause). [assumption]. 5.56/5.64 2430 (all N all Ma hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(suc,N)),Ma) = hAPP_nat_nat(nat_case_nat(zero_zero_nat,hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,suc),hAPP_nat_fun_nat_nat(ord_min_nat,N))),Ma)) # label(fact_5125_min__Suc1) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2431 (all Sa all Ta all A all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),B)) -> hBOOL(hAPP_f115998247l_bool(hAPP_P1876494581l_bool(member180897546at_nat,hAPP_P369611207at_nat(hAPP_P402336767at_nat(produc494345619at_nat,hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,A),Sa)),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,B),Ta))),pair_leq)))) # label(fact_4542_pair__leqI1) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2432 (all A_234 all C_94 all D_19 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_234),C_94)),D_19) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_234),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_94),D_19))) # label(fact_228_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2433 (all A_172 all B_147 all C_85 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_172),B_147)),C_85)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_147),C_85)))) # label(fact_1090_dvd__mult__right) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2434 (all X_45 all P_7 all Q_11 hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_45),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_11)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_45),P_7)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_45),Q_11))) # label(fact_97_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2435 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B)),zero_zero_rat)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),zero_zero_int))))) # label(fact_4715_Fract__less__zero__iff) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2436 (all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),P_3)) -> hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_2805_inv__inv__aux) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2437 (all N_1 hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_nat_real(real_nat,N_1))),pi)) = zero_zero_real) # label(fact_3676_sin__2npi) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2438 (all Z1 all Z2 all Z3 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),Z2)),Z3) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z2),Z3))) # label(fact_260_zmult__assoc) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2439 (all X_2 all N exists T hAPP_real_real(sin,X_2) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_r195310020l_real(hAPP_f2126667875l_real(cOMBC_746811033l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),zero_zero_real)),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,div_div_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,minus_minus_nat),hAPP_nat_nat(suc,zero_zero_nat)))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))))),hAPP_r474017924t_real(power_power_real,X_2))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,T),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(real_nat,N))),pi)))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),N)))) # label(fact_4922_Maclaurin__sin__expansion) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2440 (all B all A all C (C = B <-> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C),A) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B),A))) # label(fact_1936_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2441 (all M all N_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)),M))) # label(fact_1655_Nat_Odiff__le__self) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2442 (all X_1 all Y_1 (Y_1 != X_1 | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(fequal_rat,X_1),Y_1)))) # label(help_fequal_2_1_fequal_000tc__Rat__Orat_T) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2443 (all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_int_real(number267125858f_real,Y_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Y_2)))) # label(fact_757_less__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2444 (all Z_1 (is_int(Z_1) -> -(all M_1 all N_2 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(semiri1621563631at_int,M_1)),hAPP_nat_int(semiri1621563631at_int,N_2)) != Z_1))) # label(fact_3267_int__diff__cases) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2445 (all V all V_1 (is_int(V_1) & is_int(V) -> ((-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V))) -> (hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1))) -> zero_zero_int = hAPP_int_int(number_number_of_int,V)) & (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1))) -> V = V_1)) & (hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,V_1)),zero_zero_int))) <-> hAPP_int_nat(number_number_of_nat,V) = hAPP_int_nat(number_number_of_nat,V_1)))) # label(fact_3282_eq__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2446 (all B_143 all A_168 all C_82 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_168),C_82)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_168),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_143),C_82))))) # label(fact_1117_dvd__mult) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2447 (all X_4 all N_6 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_6)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_4)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_n546711566l_real(root,N_6),X_4)))))) # label(fact_3929_real__root__pos__pos) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2448 (all B_2 all A_3 (A_3 != zero_zero_nat -> (exists X exists Y hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),X) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),Y)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2))))) # label(fact_4373_bezout__nat) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2449 (all P all Q all R hAPP_f1344929775l_bool(P,hAPP_r632163725t_bool(Q,R)) = hAPP_r1150855451l_bool(hAPP_f381769257l_bool(hAPP_f512004221l_bool(cOMBB_188663909l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__HOL__034) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2450 (all A_90 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_90),one_one_real) = A_90) # label(fact_2109_mult__1__right) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2451 (all S ((exists K hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,S),hAPP_n1699378549t_bool(ord_lessThan_nat,K)))) <-> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S)))) # label(fact_4949_finite__nat__iff__bounded) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2452 (all N all P_1 (hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) -> ((all N_2 (hBOOL(hAPP_nat_bool(P_1,N_2)) -> hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,N_2))))) -> hBOOL(hAPP_nat_bool(P_1,N))))) # label(fact_3309_nat__induct) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2453 (all F all X_2 all L (hBOOL(hAPP_real_bool(deriv_real(F,X_2),L)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,L),zero_zero_real)) -> (exists D_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2)) & (all H_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H_1),D_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,X_2)),hAPP_real_real(F,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_2),H_1)))))))))))) # label(fact_3887_DERIV__neg__dec__left) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2454 (all Y_2 all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_2)) -> (zero_zero_real = X_2 & Y_2 = zero_zero_real <-> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Y_2) = zero_zero_real)))) # label(fact_2169_add__nonneg__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2455 (all A (zero_zero_rat = A <-> hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),A) = zero_zero_rat)) # label(fact_1979_double__zero__sym) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2456 (all W_1 all Z_2 (is_int(Z_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),Z_2)) <-> (exists N_2 Z_2 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_1),hAPP_nat_int(semiri1621563631at_int,N_2)))))) # label(fact_3002_zle__iff__zadd) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2457 (all B_2 all A_3 all C_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,B_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),C_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,B_2),C_1))) # label(fact_4308_gcd__left__commute__nat) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2458 (all X_1 all Y_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),hAPP_real_real(hAPP_n546711566l_real(root,N_1),Y_1)))))) # label(fact_3901_real__root__le__mono) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2459 (all A hBOOL(tendsto_complex_real(im,hAPP_complex_real(im,A),at_complex(A)))) # label(fact_5076_Im_Ocont) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2460 (all C_73 all A_154 all B_130 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_154),B_130)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_73)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_154),C_73)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_130),C_73)))))) # label(fact_1215_mult__right__mono) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2461 (all Xa_4 all X_18 all Y_14 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Xa_4),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_18),Y_14)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Xa_4),X_18)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Xa_4),Y_14))) # label(fact_2481_mult__right_Oadd) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2462 (all B_72 all C_35 all A_83 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_83)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_72),C_35)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_72),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_83),C_35)))))) # label(fact_2165_add__increasing) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2463 (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),X_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),one_one_real)) -> hBOOL(hAPP_real_bool(deriv_real(arcsin,X_2),hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))))))) # label(fact_3880_DERIV__arcsin) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2464 (all A_25 all B_22 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_25),B_22)) -> B_22 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B_22),A_25)),A_25))) # label(fact_2600_dvd__div__mult__self) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2465 (all A all E_2 all C all B all D (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),E_2)),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),E_2)),D))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B)),E_2)),C)),D)))) # label(fact_1574_le__add__iff1) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2466 (all V_3 all V_2 ((-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_3))) -> hAPP_nat_nat(div_mod_nat(hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_int_nat(nat,hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,V_3)))) & (hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_3))) -> hAPP_int_nat(number_number_of_nat,V_2) = hAPP_nat_nat(div_mod_nat(hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)))) & (hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> hAPP_nat_nat(div_mod_nat(hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = zero_zero_nat))) # label(fact_3288_mod__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2467 (all A_93 hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_93),zero_zero_rat) = A_93) # label(fact_2081_diff__0__right) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2468 (all A_166 all B_141 hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_166),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_141),A_166)))) # label(fact_1129_dvd__triv__right) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2469 (all X_1 hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,X_1)) = hAPP_complex_real(im,hAPP_complex_complex(uminus473333897omplex,X_1))) # label(fact_4099_Im_Ominus) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2470 (all X_2 (hBOOL(hAPP_int_bool(even_odd_even_int,X_2)) <-> hAPP_int_int(div_mod_int(X_2),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_int)) # label(fact_3228_even__def) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2471 (all A_175 all E_3 all B_149 all C_87 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_175),E_3)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_149),E_3)),C_87)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_175),B_149)),E_3)),C_87)) # label(fact_1070_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2472 (all C_83 all D_16 all A_169 all B_144 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_169),B_144)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,C_83),D_16)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_169),C_83)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_144),D_16)))))) # label(fact_1107_mult__dvd__mono) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2473 (all A_44 all B_37 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_44),B_37)) -> hAPP_nat_nat(div_mod_nat(B_37),A_44) = zero_zero_nat)) # label(fact_2430_dvd__imp__mod__0) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2474 (all Y_1 all X_1 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Y_1),zero_zero_int),P_3)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_1)),zero_zero_int),P_3)))))) # label(fact_1489_zcong__zmult__prop3) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2475 (all R_3 all A_3 all N_1 rcis(hAPP_nat_real(hAPP_r474017924t_real(power_power_real,R_3),N_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),A_3)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,rcis(R_3,A_3)),N_1)) # label(fact_4250_DeMoivre2) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2476 (all A_1 all B_1 (image_nat_int(semiri1621563631at_int,A_1) = image_nat_int(semiri1621563631at_int,B_1) <-> A_1 = B_1)) # label(fact_4601_transfer__nat__int__set__relations_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2477 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> hAPP_real_real(exp_real,hAPP_real_real(ln,X_1)) = X_1)) # label(fact_3846_exp__ln) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2478 (all X_2 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Y_2))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,X_2)),Y_2)))) # label(fact_3490_real__0__le__add__iff) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2479 (all M hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),M)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = M) # label(fact_2764_add__self__div__2) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2480 (all A all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A),natfloor(X_2))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,A)),X_2))))) # label(fact_3686_le__natfloor__eq) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2481 (all A all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),Ma)) -> hAPP_int_int(legacy_zgcd(big_co1548731110nt_int(cOMBI_int,hAPP_i1948725293t_bool(bnorRset(A),Ma))),Ma) = one_one_int)) # label(fact_4810_Bnor__prod__zgcd) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2482 (all K_2 all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N_1))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),M)))) # label(fact_4316_dvd__gcd__D1__nat) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2483 (all M divmod_nat(M,zero_zero_nat) = hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,zero_zero_nat),M)) # label(fact_4208_divmod__nat__zero) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2484 (all L_2 all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_2),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_2),M))))) # label(fact_1654_diff__le__mono2) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2485 (all B_95 all C_53 all A_116 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_116)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_95),C_53)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_95),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_116),C_53)))))) # label(fact_1869_add__strict__increasing2) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2486 (all N all A all B all P_1 ((all A_2 all B_4 all C_3 (hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),B_4))) -> (hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,B_4),C_3))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),B_4)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_4),C_3)) -> hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),C_3)))))))) -> (-hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A),B))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) -> -hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N))),hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N))))))))) # label(fact_4246_not__P__Bolzano__bisect) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2487 (all W_5 ((zero_zero_nat = hAPP_int_nat(number_number_of_nat,W_5) -> hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,zero_zero_complex),hAPP_int_nat(number_number_of_nat,W_5)) = one_one_complex) & (hAPP_int_nat(number_number_of_nat,W_5) != zero_zero_nat -> zero_zero_complex = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,zero_zero_complex),hAPP_int_nat(number_number_of_nat,W_5))))) # label(fact_1483_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2488 (all Z_11 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_11) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z_11),Z_11)) # label(fact_493_mult__2) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2489 (all U hAPP_nat_nat(suc,U) = hAPP_f22106695ol_nat(finite_card_nat,hAPP_n1699378549t_bool(ord_atMost_nat,U))) # label(fact_5191_card__atMost) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2490 (all A_184 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_184),zero_zero_int))) # label(fact_916_dvd__0__right) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2491 (all B all A all C (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C),A) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B),A) <-> C = B)) # label(fact_1934_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2492 (all P_1 all L ((exists X (hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,L),X))) & is_int(X))) <-> (exists X (is_int(X) & hBOOL(hAPP_int_bool(P_1,X)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,L),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),zero_zero_int))))))) # label(fact_1811_unity__coeff__ex) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2493 (all X_2 the_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_r481142414t_bool(hAPP_f1845081853t_bool(cOMBC_int_real_bool,hAPP_f801032407l_bool(hAPP_f499530369l_bool(cOMBB_1317819104ol_int,ord_less_eq_real),ring_1_of_int_real)),X_2))),hAPP_f429384717t_bool(hAPP_f1173902867t_bool(cOMBB_real_bool_int,hAPP_r1134773055l_bool(ord_less_real,X_2)),hAPP_f563293398t_real(hAPP_f134192053t_real(cOMBB_int_real_int,ring_1_of_int_real),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,plus_plus_int),one_one_int))))) = archim1246769320r_real(X_2)) # label(fact_4889_floor__real__def) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2494 (all V_16 all W_14 all Z_14 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_16)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,W_14)),Z_14)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_16),W_14))),Z_14)) # label(fact_422_mult__number__of__left) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2495 (all P all Q all R hAPP_r474017924t_real(hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,P),Q),R) = hAPP_r474017924t_real(P,hAPP_real_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__Nat__Onat_Mtc__RealDe) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2496 (all K_2 all N_1 all M hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_1)),M)),N_1) = hAPP_nat_nat(div_mod_nat(M),N_1)) # label(fact_2516_mod__mult__self3) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2497 (all V_3 all V_2 ((-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,V_3)))) & (hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3))))) # label(fact_3285_div__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2498 (all A_3 -(hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),one_one_nat)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,one_one_nat),A_3)))) # label(fact_2283_gcd__lcm__complete__lattice__nat_Onot__less__bot) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2499 (all A_226 all B_161 all V_18 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_226),hAPP_int_int(number_number_of_int,V_18))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_161),hAPP_int_int(number_number_of_int,V_18))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_226),B_161)),hAPP_int_int(number_number_of_int,V_18))) # label(fact_379_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2500 (all X_1 all Y_1 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(cnj,X_1)),hAPP_complex_complex(cnj,Y_1)) = hAPP_complex_complex(cnj,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_1),Y_1))) # label(fact_4075_cnj_Oadd) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2501 -(all S_2 (is_int(S_2) -> -(hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),S_2)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(s1),S_2),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,S_2),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)))))) # label(fact_544__096_B_Bthesis_O_A_I_B_Bs_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2502 (all A_60 all B_52 all C_23 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(div_mod_int(A_60),C_23)),hAPP_int_int(div_mod_int(B_52),C_23))),C_23) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_60),B_52)),C_23)) # label(fact_2358_mod__add__eq) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2503 (all X_1 all Y_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),X_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_1),Y_1)))))) # label(fact_3336_real__sqrt__mult__self__sum__ge__zero) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2504 (all N_1 one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(suc,N_1)),N_1)) # label(fact_4346_coprime__Suc__nat) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2505 (all A_170 all B_145 all K_7 (hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_145),K_7) = A_170 -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_145),A_170)))) # label(fact_1104_dvdI) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2506 (all Ma all K_1 all N (N = Ma | K_1 = zero_zero_nat <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),K_1))) # label(fact_1757_mult__cancel2) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2507 (all X_2 all K_1 (hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),K_1) = X_2 -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven)))) # label(fact_3225_zEvenI) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2508 (all A_73 zero_zero_int = hAPP_int_int(div_mod_int(zero_zero_int),A_73)) # label(fact_2310_mod__0) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2509 (all U all Ma all N all J all I_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),I_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),J)),U)),Ma)),N)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U)),N)))))) # label(fact_1839_nat__less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2510 (all A_22 all C_7 all B_19 (B_19 != zero_z891286103de_int -> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_7),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_22),B_19)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_22),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_19),C_7))),B_19))) # label(fact_2646_div__mult__self2) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2511 (all N_28 all K_5 all M_15 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,K_5),M_15)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,K_5),N_28)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,K_5),hAPP_Q1762011733de_int(div_mo231679042de_int(M_15),N_28)))))) # label(fact_2383_dvd__mod) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2512 (all B_90 all A_111 all C_50 (is_int(C_50) & is_int(B_90) -> (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_50),A_111) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_90),A_111) -> C_50 = B_90))) # label(fact_1911_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2513 (all F all K_1 all A all B (B != A -> ((all X hBOOL(hAPP_real_bool(deriv_real(F,X),K_1))) -> K_1 = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,B)),hAPP_real_real(F,A))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B),A))))) # label(fact_3890_DERIV__const__ratio__const2) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2514 (all Y_22 all X_28 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),X_28)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Y_22)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_22),one_one_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_22),X_28)),X_28)))))) # label(fact_1469_mult__left__le__one__le) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2515 (all P all Q all R hAPP_i1948725293t_bool(hAPP_f428220345t_bool(hAPP_f654702867t_bool(cOMBB_591320580ol_int,P),Q),R) = hAPP_f1805168059t_bool(P,hAPP_i1948725293t_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc_) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2516 (all X_2 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2)) -> hBOOL(hAPP_real_bool(isCont_real_real(hAPP_n546711566l_real(root,N)),X_2))))) # label(fact_5169_isCont__root__pos) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2517 (all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2)) -> (A_3 != B_2 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),A_3))))) # label(fact_1699_dvd_Ole__neq__trans) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2518 (all N_55 all A_205 all B_156 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_205),B_156)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_205)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_205),N_55)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_156),N_55)))))) # label(fact_655_power__mono) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2519 (all K_1 hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),K_1))))) # label(fact_4730_finite__Collect__less__nat) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2520 (all W_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,W_1)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),zero_zero_int)))) # label(fact_714_bin__less__0__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2521 (all F all K_1 all A all B (B != A -> ((all X hBOOL(hAPP_real_bool(deriv_real(F,X),K_1))) -> hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,B)),hAPP_real_real(F,A)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B),A)),K_1)))) # label(fact_3891_DERIV__const__ratio__const) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2522 (all N hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f451358433l_real(hAPP_f335634176l_real(cOMBS_337378985l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,div_div_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),zero_zero_real))),hAPP_r474017924t_real(power_power_real,zero_zero_real))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(suc,N))) = one_one_real) # label(fact_4912_sumr__cos__zero__one) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2523 (all P all Q all R hAPP_P621635040nt_int(P,hAPP_i1524277240nt_int(Q,R)) = hAPP_i1153444985nt_int(hAPP_f1148171384nt_int(hAPP_f334321603nt_int(cOMBB_1484610687nt_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc_) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2524 (all B_95 all C_53 all A_116 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_116)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_95),C_53)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_95),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_116),C_53)))))) # label(fact_1868_add__strict__increasing2) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2525 (all C all B all A (zero_z126310315umeral != A -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A),B)) -> (hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,B),A) = C <-> hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C),A) = B)))) # label(fact_2663_dvd__div__eq__mult) # label(axiom) # label(non_clause). [assumption]. 5.56/5.64 2526 (all A all C all B (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A),C)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B),C))) <-> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A),B)) | zero_zero_complex = C)) # label(fact_1379_dvd__mult__cancel__right) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2527 (all P all Q all R hAPP_int_int(hAPP_nat_fun_int_int(hAPP_f2105620693nt_int(cOMBC_int_nat_int,P),Q),R) = hAPP_nat_int(hAPP_int_fun_nat_int(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Nat__Onat_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2528 (all D all Sa exists Z all X (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z),X)) -> (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Sa))) <-> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Sa)))))) # label(fact_2262_pinf_I10_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2529 (all A_219 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_219),A_219)),A_219) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_219),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) # label(fact_481_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2530 (all L_2 hAPP_int_int(bit0,L_2) != min) # label(fact_930_rel__simps_I42_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2531 (all D all Sa exists Z all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Z)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Sa))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Sa)))))) # label(fact_2252_minf_I9_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2532 (all K_2 all L_2 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(bit1,K_2)),hAPP_int_int(bit0,L_2)) = hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),L_2))) # label(fact_1563_diff__bin__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2533 (all W_17 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,W_17)),hAPP_int_int(number_number_of_int,W_17)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,W_17)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_17_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2534 (all U_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),U_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,U_1),hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),U_1)))) # label(fact_3363_lemma__real__divide__sqrt__less) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2535 (all N_13 all A_7 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_7)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_7),hAPP_nat_nat(suc,N_13)))))) # label(fact_3121_power__gt1) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2536 (all X_2 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(number_number_of_rat,X_2)),zero_zero_rat)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),pls)))) # label(fact_767_le__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2537 (all I all K_2 all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_2)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J_2)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_2)))) # label(fact_1671_diff__add__assoc) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2538 (all A_239 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_239),A_239) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_239),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_38_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2539 (all C_64 all D_11 all A_130 all B_109 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_130),B_109)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_64),D_11)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_130)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_64)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_130),C_64)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_109),D_11)))))))) # label(fact_1421_mult__less__le__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2540 (all X_1 all Y_1 (X_1 = Y_1 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)))) # label(fact_1696_dvd_Oeq__refl) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2541 (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),one_one_real)) -> hBOOL(monoseq_real(hAPP_r474017924t_real(power_power_real,X_2)))))) # label(fact_4805_monoseq__realpow) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2542 (all X_9 all Q_7 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_9),Q_7)),X_9) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_9),hAPP_nat_nat(suc,Q_7))) # label(fact_3012_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2543 (all P_3 all Q_2 (hBOOL(hAPP_nat_bool(prime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_3),Q_2))) -> one_one_nat = Q_2 | one_one_nat = P_3)) # label(fact_4171_prime__product) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2544 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> zero_zero_real != hAPP_real_real(sqrt,X_1))) # label(fact_3327_real__sqrt__not__eq__zero) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2545 (all A_19 all B_16 (zero_z891286103de_int != B_16 -> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_19),B_16)),one_on1684967323de_int) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_16),A_19)),B_16))) # label(fact_2658_div__add__self1) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2546 (all A_90 A_90 = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_90),one_one_complex)) # label(fact_2111_mult__1__right) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2547 (all A_3 ((A_3 = zero_zero_real -> zero_zero_real = hAPP_real_real(sgn_sgn_real,A_3)) & (A_3 != zero_zero_real -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> one_one_real = hAPP_real_real(sgn_sgn_real,A_3)) & (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> hAPP_real_real(sgn_sgn_real,A_3) = hAPP_real_real(uminus_uminus_real,one_one_real))))) # label(fact_3574_sgn__real__def) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2548 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(ln,X_1))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_1))))) # label(fact_3439_ln__gt__zero__imp__gt__one) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2549 (all Z_1 all X_1 all Y_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y_1),Z_1))))) # label(fact_4156_termination__basic__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2550 (all A_237 all B_166 all C_97 all D_20 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_237),B_166)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_97),D_20)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_237),C_97)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_166),D_20))) # label(fact_194_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2551 (all X hAPP_nat_real(diffs_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))),X) = hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,X)))) # label(fact_4875_exp__fdiffs) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2552 (all C_79 all D_15 all A_160 all B_136 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_160),B_136)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_79),D_15)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_136)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_79)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_160),C_79)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_136),D_15)))))))) # label(fact_1184_mult__mono) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2553 (all A_208 A_208 = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,zero_z126310315umeral),A_208)) # label(fact_622_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2554 (all A_11 all N_18 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_11),N_18)),A_11) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_11),hAPP_nat_nat(suc,N_18))) # label(fact_3032_power__Suc2) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2555 (all Lx_4 all Ly_2 all Rx_4 all Ry_2 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_4),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Ly_2),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_4),Ry_2))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_4),Ly_2)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_4),Ry_2))) # label(fact_156_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2556 (all C_89 all D_17 all A_200 all B_154 all R_8 (is_int(R_8) & is_int(D_17) & is_int(C_89) -> (R_8 != zero_zero_int -> (C_89 != D_17 & B_154 = A_200 -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_154),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_8),D_17)) != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_200),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_8),C_89)))))) # label(fact_693_add__scale__eq__noteq) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2557 (all A_177 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_177),zero_zero_int) = zero_zero_int) # label(fact_1036_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2558 (all Y_2 all X_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat,X_2)),hAPP_int_nat(nat,Y_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Y_2)))))) # label(fact_2751_transfer__nat__int__relations_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2559 (all C all A all B (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A),B)) | C = zero_zero_complex <-> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,C),A)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,C),B))))) # label(fact_1375_dvd__mult__cancel__left) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2560 (all U all Ma all N all J all I_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),I_1)) -> (N = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),J)),U)),Ma) <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U)),N)))) # label(fact_1850_nat__eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2561 (all N_67 all A_218 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_218)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_218),N_67)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_218),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_218),N_67)))))) # label(fact_517_power__less__power__Suc) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2562 (all A all B all P_1 ((all A_2 all B_4 all C_3 (hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,B_4),C_3))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_4),C_3)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),B_4)) & hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),B_4))) -> hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),C_3))))) -> (-hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A),B))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) -> (all N_2 -hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_2))),hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_2)))))))))) # label(fact_4247_not__P__Bolzano__bisect_H) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2563 (all P all Q all R hAPP_P1668131738nt_int(hAPP_f167782601nt_int(hAPP_f1769710159nt_int(cOMBC_2037038898nt_int,P),Q),R) = hAPP_f521271395nt_int(hAPP_P752050971nt_int(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc_) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2564 (all A_76 all B_65 hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(div_mo1740067990umeral(A_76),B_65)),B_65) = hAPP_C498520661umeral(div_mo1740067990umeral(A_76),B_65)) # label(fact_2300_mod__mod__trivial) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2565 (all A_3 all B_2 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),one_one_int))) # label(fact_1159_zcong__1) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2566 (all I_1 all J (is_int(J) & is_int(I_1) -> (zero_zero_int = hAPP_int_int(legacy_zgcd(I_1),J) <-> J = zero_zero_int & zero_zero_int = I_1))) # label(fact_4487_zgcd0) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2567 (all X_2 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),one_one_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X_2)),one_one_real))))) # label(fact_3922_real__root__le__1__iff) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2568 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) -> (exists K hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K)) = N_1))) # label(fact_3261_less__imp__Suc__add) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2569 (all J_2 all A_3 all P_3 all K_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),K_2))),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,J_2),K_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),K_2))),K_2)),P_3)))) # label(fact_1895_aux______1) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2570 (all A_210 zero_zero_rat = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,zero_zero_rat),A_210)) # label(fact_614_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2571 (exists K_4 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),K_4)) & (all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(im,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X)),K_4)))))) # label(fact_4107_Im_Ononneg__bounded) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2572 (all B_2 hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_2),one_one_int)) = negDivAlg(hAPP_int_int(number_number_of_int,min),B_2)) # label(fact_2494_negDivAlg__minus1) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2573 (all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(arccos,Y_1)))))) # label(fact_3605_arccos__lbound) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2574 (all K_1 hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_atMost_nat,K_1)))) # label(fact_5184_finite__atMost) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2575 (all X_19 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_19),zero_zero_complex) = zero_zero_complex) # label(fact_2477_mult__right_Ozero) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2576 (all X_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),hAPP_int_int(bit1,pls))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,X_2)),one_one_real)))) # label(fact_291_le__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2577 (all Z_7 all W_2 all Y_10 all X_14 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,Y_10),X_14)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,Z_7),W_2)) -> hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,X_14),Y_10)),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,W_2),Z_7)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,X_14),W_2)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Y_10),Z_7))))) # label(fact_2605_div__mult__div__if__dvd) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2578 (all A_110 all B_89 all C_49 (hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_110),C_49) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_110),B_89) -> C_49 = B_89)) # label(fact_1922_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2579 (all P all Q all R hAPP_r373117745t_bool(hAPP_f1459420095t_bool(hAPP_f693283245t_bool(cOMBC_1113900202t_bool,P),Q),R) = hAPP_f1222178131t_bool(hAPP_r2052264067t_bool(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Int__Oint_Mtc__fun_It_028) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2580 (all X_2 all Y_2 (is_int(Y_2) & is_int(X_2) -> (X_2 = Y_2 <-> hAPP_int_rat(number_number_of_rat,Y_2) = hAPP_int_rat(number_number_of_rat,X_2)))) # label(fact_250_eq__number__of) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2581 (all C_81 all A_167 all B_142 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_167),B_142)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_167),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_142),C_81))))) # label(fact_1125_dvd__mult2) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2582 (all X_2 (hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_nat_int(semiri1621563631at_int,X_2))) <-> hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2)))) # label(fact_3764_Parity_Otransfer__int__nat__relations) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2583 (all A_23 all B_20 zero_z126310315umeral = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_23),B_20)),B_20)) # label(fact_2615_mod__div__trivial) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2584 (all A_198 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_198),one_one_nat) = A_198) # label(fact_752_power__one__right) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2585 (all B_113 all A_136 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_136)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_113)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_136),B_113)))))) # label(fact_1319_mult__pos__pos) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2586 (all M_7 all N_21 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(semiri1619134803umeral,M_7)),hAPP_n1108039445umeral(semiri1619134803umeral,N_21)) = hAPP_n1108039445umeral(semiri1619134803umeral,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_7),N_21))) # label(fact_2962_of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2587 (all M hAPP_int_int(legacy_zgcd(one_one_int),M) = one_one_int) # label(fact_4486_zgcd__1__left) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2588 (all Ma all N divmod_nat(Ma,N) = the_Pr588456374at_nat(divmod_nat_rel(Ma,N))) # label(fact_4888_divmod__nat__def) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2589 (all Ma (one_one_nat = Ma <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),one_one_nat)))) # label(fact_1786_nat__dvd__1__iff__1) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2590 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),pi)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sin,X_1)))))) # label(fact_3656_sin__ge__zero) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2591 (all A_3 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))) -> hAPP_int_int(inv(P_3),A_3) != zero_zero_int)))) # label(fact_2142_inv__not__0) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2592 (all X_6 all Y_5 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_6),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_5),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_6),Y_5)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_6),Y_5))) # label(fact_3193_realpow__two__diff) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2593 (all K_1 all F all N (is_int(K_1) -> ((all I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),N)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_2),one_one_nat))),hAPP_nat_int(F,I_2)))),one_one_int)))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,zero_zero_nat)),K_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),hAPP_nat_int(F,N))) -> (exists I_2 (K_1 = hAPP_nat_int(F,I_2) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),N))))))))) # label(fact_3477_nat0__intermed__int__val) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2594 (all K_1 all I_1 all J hAPP_n1699378549t_bool(ord_at4362885an_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),K_1)) = image_nat_nat(hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),K_1),hAPP_n1699378549t_bool(ord_at4362885an_nat(I_1),J))) # label(fact_4905_image__add__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2595 (all B_143 all A_168 all C_82 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_168),C_82)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_168),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_143),C_82))))) # label(fact_1119_dvd__mult) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2596 (all Ma all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,Ma)),hAPP_nat_nat(semiri984289939at_nat,N))))) # label(fact_2935_of__nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2597 (all X_7 all Q_5 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_7),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_7),Q_5)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_7),hAPP_nat_nat(suc,Q_5))) # label(fact_3024_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2598 (all A_14 all B_11 all C_4 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_14),C_4) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_14),B_11)),B_11)),hAPP_C498520661umeral(div_mo1740067990umeral(A_14),B_11))),C_4)) # label(fact_2685_div__mod__equality) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2599 (all N_1 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,M),N_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),hAPP_int_int(fact_fact_int,N_1)))))) # label(fact_4283_dvd__fact__int) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2600 (all A_3 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),hAPP_nat_nat(suc,zero_zero_nat)))) # label(fact_4022_coprime__Suc0) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2601 (all A_228 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_228),one_on1645066479umeral) = A_228) # label(fact_330_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2602 (all Ma all N (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,Ma)),hAPP_int_real(number267125858f_real,N))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ma),hAPP_int_int(number_number_of_int,N))))) # label(fact_4443_number__of__le__real__of__int__iff2) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2603 (all Z_1 all N_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_1),N_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),N_1))))) # label(fact_601_zdvd__imp__le) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2604 (all N_1 N_1 = natfloor(hAPP_nat_real(real_nat,N_1))) # label(fact_3637_natfloor__real__of__nat) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2605 (all F all L all C (hBOOL(tendsto_real_real(F,L,at_real(C))) -> (zero_zero_real != L -> (exists R_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_1)) & (all X (X != C & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),X))),R_1)) -> hAPP_real_real(F,X) != zero_zero_real))))))) # label(fact_5087_LIM__fun__not__zero) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2606 t = one_one_int -> (exists X exists Y (is_int(X) & is_int(Y) & hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))) # label(fact_1__096t_A_061_A1_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_06) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2607 (all Ma all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_nat_nat(suc,N))))) # label(fact_3065_less__Suc__eq__le) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2608 (all Ma all K_1 all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),N)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),hAPP_int_int(div_mod_int(Ma),N))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),Ma))))) # label(fact_2392_dvd__mod__iff) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2609 (all X_2 all Y_2 (X_2 = zero_zero_real & zero_zero_real = Y_2 <-> zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_802_sum__power2__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2610 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X_1))),X_1)))) # label(fact_3454_ln__add__one__self__le__self) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2611 (all Lx_2 all Ly all Rx_2 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ly),Rx_2)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_2),Ly)),Rx_2)) # label(fact_167_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2612 (all A_30 all B_26 all C_11 (zero_zero_nat != C_11 -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_30),B_26) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_30),C_11)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_26),C_11)))) # label(fact_2578_div__mult__mult2) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2613 (all A_215 all M_24 all N_64 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_24),N_64)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_215),M_24)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_215),N_64))))) # label(fact_558_le__imp__power__dvd) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2614 (all Q_2 all B_2 all R_3 all C_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(div_mod_int(Q_2),C_1))),R_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),C_1))))))) # label(fact_2464_zmult2__lemma__aux4) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2615 (all X_2 all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,X_2)),hAPP_nat_int(semiri1621563631at_int,Y_2))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),Y_2)))) # label(fact_3000_Nat__Transfer_Otransfer__int__nat__relations_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2616 (all A_10 all N_17 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_10),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_10),N_17)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_10),hAPP_nat_nat(suc,N_17))) # label(fact_3040_power__Suc) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2617 (all X_1 all Y_1 hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_1) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),hAPP_int_int(abs_abs_int,Y_1))) # label(fact_4990_gcd__abs2__int) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2618 (all K_1 all N_2 all F_2 hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F_2),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_2),K_1))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,F_2),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),K_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_2))),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F_2),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),K_1)))) # label(fact_4898_sumr__offset4) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2619 (all M_22 all A_214 all N_62 all B_159 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_214),N_62)),B_159)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_22),N_62)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_214),M_22)),B_159))))) # label(fact_572_power__le__dvd) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2620 (all B_1_1 is_bool(vanishes(B_1_1))) # label(gsy_c_RealDef_Ovanishes) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2621 (all C_73 all A_154 all B_130 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_154),B_130)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_73)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_154),C_73)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_130),C_73)))))) # label(fact_1216_mult__right__mono) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2622 (all B_92 all A_113 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_113)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_92)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_113),B_92)))))) # label(fact_1888_add__pos__nonneg) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2623 (all F all Y_2 all G all X_2 (hBOOL(hAPP_real_bool(sums_real(G),X_2)) -> (hBOOL(hAPP_real_bool(sums_real(F),Y_2)) -> hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f335634176l_real(cOMBS_337378985l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,F),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,div_div_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,G),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,div_div_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,minus_minus_nat),one_one_nat))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Y_2)))))) # label(fact_4868_sums__if) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2624 (all A all E_2 all C all B all D (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B),A)),E_2)),D) = C <-> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),E_2)),C) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),E_2)),D))) # label(fact_1548_eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2625 (all X_11 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_11)) -> hAPP_nat_complex(semiri2020571505omplex,hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_11))) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_nat_complex(semiri2020571505omplex,hAPP_int_nat(nat,X_11))),hAPP_nat_complex(semiri2020571505omplex,hAPP_int_nat(nat,X_11))))) # label(fact_2813_of__nat__double) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2626 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N_1) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)),N_1))) & (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) -> zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N_1)))) # label(fact_3185_div__if) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2627 (all W_1 all Z_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat,W_1)),hAPP_int_nat(nat,Z_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2))))) # label(fact_2740_nat__mono__iff) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2628 (all A_1 A_1 = image_int_nat(nat,image_nat_int(semiri1621563631at_int,A_1))) # label(fact_4603_transfer__int__nat__set__return__embed) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2629 (all A_231 all N_68 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_231),N_68)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_231),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_68))) # label(fact_282_power__even__eq) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2630 (all L_2 all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),L_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),L_2))))) # label(fact_1653_diff__le__mono) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2631 (all A_3 all B_2 all C_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1)) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),C_1))),hAPP_int_int(div_mod_int(A_3),B_2)) = hAPP_int_int(div_mod_int(A_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),C_1)))) # label(fact_2696_zmod__zmult2__eq) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2632 (all B_69 all A_80 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_80)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_69)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_80),B_69)))))) # label(fact_2193_add__pos__pos) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2633 (all A_229 all B_162 all C_91 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_229),B_162)),C_91) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_229),C_91)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_162),C_91))) # label(fact_313_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2634 (all N_9 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(semiri151668891at_rat,N_9)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_9)))) # label(fact_3215_of__nat__less__two__power) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2635 (all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K_1)),min)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),min)))) # label(fact_945_rel__simps_I28_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2636 (all A hBOOL(hAPP_complex_bool(isCont_complex_real(im),A))) # label(fact_5165_Im_OisCont) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2637 (all N_1 N_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N_1)),one_one_nat)) # label(fact_3075_diff__Suc__1) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2638 (all A_220 (is_int(A_220) -> A_220 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_220),hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,pls))))) # label(fact_456_mult__numeral__1__right) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2639 (all X_1 all N_1 all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> hAPP_real_real(hAPP_n546711566l_real(root,M),hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)) = hAPP_real_real(hAPP_n546711566l_real(root,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1)),X_1))))) # label(fact_3931_real__root__pos__mult__exp) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2640 (all X_2 all Y_2 (zero_zero_rat = X_2 & Y_2 = zero_zero_rat <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),zero_zero_rat)))) # label(fact_850_sum__power2__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2641 (all X_1 all M hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,M))),pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,M)),pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_3706_cos__expansion__lemma) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2642 (all A_196 -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_196),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),zero_zero_real))) # label(fact_796_power2__less__0) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2643 (all W hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,W)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,W)),hAPP_int_nat(number_number_of_nat,W))) # label(fact_4746_nat__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2644 (all P all Q all R hAPP_int_bool(hAPP_f1448486952t_bool(hAPP_f696844121t_bool(cOMBB_nat_bool_int,P),Q),R) = hAPP_nat_bool(P,hAPP_int_nat(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__HOL__Obool_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2645 (all K_1 all Xa_1 hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,cos),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,plus_plus_real),K_1)),Xa_1),hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Xa_1),K_1)))))) # label(fact_4739_DERIV__cos__add) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2646 (all A_128 all C_62 all B_107 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_128),C_62)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_107),C_62))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_62)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_128),B_107))))) # label(fact_1432_mult__right__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2647 (all A_3 all B_2 all C_1 (is_int(C_1) -> (zero_zero_int != C_1 -> hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_1),A_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_1),B_2))))) # label(fact_4685_mult__rat__cancel) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2648 (all B_114 all A_137 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_137)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_114),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_137),B_114)),zero_zero_real))))) # label(fact_1310_mult__pos__neg) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2649 (all M_27 all A_224 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_224),one_one_real)),M_27) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,M_27),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_224),M_27))) # label(fact_401_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2650 (all A all P_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)),P_5)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int))),one_one_int),P_5)))) # label(fact_1590_inv__not__p__minus__1__aux) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2651 (all W all Z_1 hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z_1))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,W))),hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,Z_1)))) # label(fact_3435_nat__abs__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2652 (all A all C all B (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A),B)) <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A),C)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B),C))))) # label(fact_2034_add__le__cancel__right) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2653 (all A_235 all B_164 all C_95 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_235),B_164)),C_95) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_235),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_164),C_95))) # label(fact_225_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2654 (all A_3 all B_2 all C_1 all D_3 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)),hAPP_int_rat(hAPP_int_fun_int_rat(fract,C_1),D_3)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),C_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),D_3))) # label(fact_4689_mult__rat) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2655 (all A_41 all A_40 all B_33 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_41),B_33)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_40),B_33)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_41),A_40)),B_33)) # label(fact_2482_mult_Oadd__left) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2656 (all X_1 (-hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1)) -> hAPP_nat_nat(suc,zero_zero_nat) = hAPP_nat_nat(div_mod_nat(X_1),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))))) # label(fact_3772_odd__nat__mod__two__eq__one) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2657 (all A_46 all B_39 all C_14 hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_46),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_39),C_14))),B_39) = hAPP_nat_nat(div_mod_nat(A_46),B_39)) # label(fact_2420_mod__mult__self2) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2658 (all P all Q all R hAPP_nat_complex(hAPP_f219720582omplex(hAPP_f1052840010omplex(cOMBS_696648398omplex,P),Q),R) = hAPP_real_complex(hAPP_n1983812447omplex(P,R),hAPP_nat_real(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__Complex__Ocom) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2659 (all X_1 all Y_1 (is_int(Y_1) -> hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,fFalse),X_1),Y_1) = Y_1)) # label(help_If_2_1_If_000tc__Int__Oint_T) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2660 (all R_3 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,archim856651990g_real(R_3))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,R_3),one_one_real)))) # label(fact_4456_real__of__int__ceiling__le__add__one) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2661 (all Z_1 (is_int(Z_1) -> ((all N_2 Z_1 != hAPP_nat_int(semiri1621563631at_int,N_2)) -> -(all N_2 hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_2))) != Z_1)))) # label(fact_4546_int__cases) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2662 (all A_195 all N_47 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_195),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_195),N_47)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_195),N_47)),A_195)) # label(fact_809_power__commutes) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2663 (all K_2 zero_zero_rat = hAPP_int_rat(hAPP_int_fun_int_rat(fract,K_2),zero_zero_int)) # label(fact_4672_rat__number__collapse_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2664 (all A_86 all B_75 A_86 = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_86),B_75)),B_75)) # label(fact_2140_diff__add__cancel) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2665 (all C_68 all A_143 all B_120 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_143),B_120)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_68)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_143),C_68)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_120),C_68)))))) # label(fact_1275_mult__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2666 (all A_10 all N_17 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_10),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_10),N_17)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_10),hAPP_nat_nat(suc,N_17))) # label(fact_3038_power__Suc) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2667 (all P all Q all R hAPP_real_real(hAPP_f2119584768l_real(hAPP_f59594445l_real(cOMBC_2010481033l_real,P),Q),R) = hAPP_f1406902706l_real(hAPP_r1409264866l_real(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__O) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2668 (all B_1 all K_1 all Diff all H all N ((all M_1 all T (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),H)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),T)) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),T),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T))))) -> (N = hAPP_nat_nat(suc,K_1) -> (all M_1 all T (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),H)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),T)) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,minus_minus_real),hAPP_n546711566l_real(Diff,M_1))),hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,plus_plus_real),hAPP_f2119584768l_real(hAPP_f59594445l_real(cOMBC_2010481033l_real,hAPP_f1523861863l_real(hAPP_f1067681227l_real(cOMBB_1934313333l_real,big_co604158596t_real),hAPP_f1993765077t_real(hAPP_f895854793t_real(cOMBB_1948233853l_real,hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f1904307534l_real(hAPP_f224989205l_real(cOMBB_1419615765al_nat,Diff),hAPP_nat_fun_nat_nat(plus_plus_nat,M_1))),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))))),power_power_real))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),M_1))))),hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,hAPP_r1250527377l_real(times_times_real,B_1)),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,inverse_divide_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,power_power_real),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),M_1)))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),M_1))))))),T),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f1904307534l_real(hAPP_f224989205l_real(cOMBB_1419615765al_nat,Diff),hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(suc,M_1)))),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,T))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,M_1))))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,T),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,M_1)))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,M_1))))))))))))))) # label(fact_4923_Maclaurin__lemma2) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2669 (all B_69 all A_80 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_80)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_69)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_80),B_69)))))) # label(fact_2191_add__pos__pos) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2670 (all A all C all D all B (is_int(B) & is_int(D) -> (B != zero_zero_int -> (zero_zero_int != D -> (hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),D) <-> hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,C),D)))))) # label(fact_4686_eq__rat_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2671 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),A_3)) -> zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)))) # label(fact_2624_div__neg__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2672 (all A_210 zero_z891286103de_int = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,zero_z891286103de_int),A_210)) # label(fact_613_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2673 (all X_45 all P_7 all Q_11 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_45),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_11)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_45),P_7)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_45),Q_11))) # label(fact_103_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2674 (all Y_3 all B all X_3 all A (hBOOL(tendsto_nat_real(X_3,A,sequentially)) -> (hBOOL(tendsto_nat_real(Y_3,B,sequentially)) -> hBOOL(tendsto_nat_complex(hAPP_f219720582omplex(hAPP_f1052840010omplex(cOMBS_696648398omplex,hAPP_f23851874omplex(hAPP_f406530075omplex(cOMBB_1714920583ex_nat,complex),X_3)),Y_3),hAPP_real_complex(hAPP_r265291036omplex(complex,A),B),sequentially))))) # label(fact_5057_LIMSEQ__Complex) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2675 (all W_1 all Z_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat,W_1)),hAPP_int_nat(nat,Z_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_2)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2)))) # label(fact_2741_zless__nat__conj) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2676 (all W_1 all Z_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_2),one_one_int))))) # label(fact_1586_zle__diff1__eq) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2677 (all D all Sa exists Z all X (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Z)) -> (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Sa))) <-> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Sa)))))) # label(fact_2268_minf_I10_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2678 (all X_1 all Y_1 all Z_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y_1),Z_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_1),Z_1))) # label(fact_1722_nat__add__left__commute) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2679 (all X_1 all Q_2 all N_1 all R_3 (X_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Q_2),N_1)),R_3) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),R_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,R_3),N_1)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N_1),X_1)))))) # label(fact_1624_divides__div__not) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2680 (all P all Q all R hAPP_nat_fun_int_int(hAPP_f1935805932nt_int(hAPP_f125788821nt_int(cOMBB_859312247nt_nat,P),Q),R) = hAPP_int_fun_int_int(P,hAPP_nat_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_001) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2681 (all Ta all B_1 all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1)) -> (all X (is_int(X) -> ((all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B_1)) -> X != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa))))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X),Ta)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),D_1)),Ta)))))))) # label(fact_5091_bset_I6_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2682 (all M_17 all N_30 ((N_30 != zero_zero_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_17),N_30) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_30),one_one_nat)))) & (N_30 = zero_zero_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_17),N_30) = one_one_nat))) # label(fact_1830_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2683 (all N_1 all A_3 all B_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),N_1)) -> (hAPP_int_int(legacy_zgcd(M),N_1) = one_one_int -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N_1))))))) # label(fact_4514_zcong__zgcd__zmult__zmod) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2684 (all C_79 all D_15 all A_160 all B_136 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_160),B_136)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,C_79),D_15)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_136)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_79)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_160),C_79)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_136),D_15)))))))) # label(fact_1185_mult__mono) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2685 (all X_2 all Y_2 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_2)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2)) <-> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_2)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2)))) # label(fact_1694_dvd_Oless__le__not__le) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2686 (all X_3 (hBOOL(cauchy_complex(X_3)) -> hBOOL(cauchy_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,re),X_3))))) # label(fact_4757_Re_OCauchy) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2687 (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(exp_real,X_2)),one_one_real)))) # label(fact_3842_exp__less__one__iff) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2688 (all B_1_1 is_bool(bseq_real(B_1_1))) # label(gsy_c_SEQ_OBseq_000tc__RealDef__Oreal) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2689 (all A_92 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_92),A_92) = zero_zero_int) # label(fact_2082_diff__self) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2690 (all P all Q all R hAPP_r1962736578t_bool(hAPP_f1063001928t_bool(hAPP_f2050994651t_bool(cOMBB_2138578194l_real,P),Q),R) = hAPP_f1545556668t_bool(P,hAPP_r1150855451l_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__fun_) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2691 (all X_2 all Y_2 all B (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),B)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),Y_2)) <-> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B),X_2)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B),Y_2)))))) # label(fact_881_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2692 (all M hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(fact_fact_int,M)))) # label(fact_4276_fact__ge__zero__int) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2693 (all A_192 all M_20 all N_43 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_192),M_20)),N_43) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_192),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_20),N_43))) # label(fact_835_power__mult) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2694 (all I_1 all K_1 all J (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),J))))) # label(fact_1674_le__diff__conv2) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2695 (all N_33 one_one_real = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_33))) # label(fact_1170_power__m1__even) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2696 (all A_3 zero_zero_int = hAPP_int_int(div_mod_int(A_3),hAPP_int_int(number_number_of_int,min))) # label(fact_2456_zmod__minus1__right) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2697 (all L all U hAPP_i1948725293t_bool(ord_at641636577an_int(L),U) = image_int_int(hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,plus_plus_int),L),hAPP_i1948725293t_bool(ord_at641636577an_int(zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,U),L)))) # label(fact_4921_image__add__int__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2698 (all X_3 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma)) -> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,image_int_int(standardRes(Ma),X_3)),collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),Ma))))))) # label(fact_4786_SR__pos) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2699 (all Z_1 (is_int(Z_1) -> Z_1 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),Z_1))) # label(fact_646_zadd__0) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2700 (all R_3 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,archim1246769320r_real(R_3))),R_3))) # label(fact_4400_real__of__int__floor__le) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2701 (all Qr_2 all M all N_1 all Qr_1 (hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N_1),Qr_1)) -> (hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N_1),Qr_2)) -> Qr_1 = Qr_2))) # label(fact_4214_divmod__nat__rel__unique) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2702 (all N_58 ((zero_zero_nat = N_58 -> one_on1645066479umeral = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,zero_z126310315umeral),N_58)) & (N_58 != zero_zero_nat -> zero_z126310315umeral = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,zero_z126310315umeral),N_58)))) # label(fact_597_power__0__left) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2703 (all B_115 all A_138 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_138)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_115),zero_zero_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_115),A_138)),zero_zero_nat))))) # label(fact_1303_mult__pos__neg2) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2704 (all P all Q all R hAPP_int_bool(hAPP_i1948725293t_bool(P,R),Q) = hAPP_int_bool(hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Int__Oint_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2705 (all B_125 all A_149 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_149)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_125)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_149),B_125)))))) # label(fact_1244_mult__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2706 (all X_2 all N all Y_2 (hAPP_nat_nat(div_mod_nat(Y_2),N) = hAPP_nat_nat(div_mod_nat(X_2),N) <-> (exists Q1 exists Q2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),Q2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),Q1))))) # label(fact_2535_nat__mod__eq__iff) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2707 (all X_1 all Y_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y_1),hAPP_nat_nat(div_mod_nat(X_1),Y_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Y_1)) # label(fact_4341_gcd__red__nat) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2708 (all K_2 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(suc,zero_zero_nat)),K_2))) # label(fact_2822_dvd__1__left) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2709 (all A all P_5 image_275383677t_bool(multInvPair(A,P_5),hAPP_i1948725293t_bool(sRStar,P_5)) = setS(A,P_5)) # label(fact_4606_SetS__def) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2710 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) -> N_1 != zero_zero_nat)) # label(fact_1743_gr__implies__not0) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2711 (all X_1 all A_3 all B_2 hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_3),B_2)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),B_2))) # label(fact_4411_powr__add) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2712 (all K_2 all M all N_1 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,K_2)),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N_1)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N_1))) # label(fact_4996_gcd__mult__distrib__int) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2713 (all M_10 all N_24 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_10),N_24)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,M_10)),hAPP_nat_real(semiri132038758t_real,N_24))))) # label(fact_2940_less__imp__of__nat__less) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2714 (all K_1 all L (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K_1)),hAPP_int_int(bit0,L))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),L)))) # label(fact_117_rel__simps_I31_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2715 (all V_22 all V_21 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_21)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_22)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_21)),hAPP_int_nat(number_number_of_nat,V_22)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_21),V_22))))) # label(fact_368_semiring__mult__number__of) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2716 (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),one_one_real)) -> hBOOL(tendsto_nat_real(hAPP_r474017924t_real(power_power_real,X_2),zero_zero_real,sequentially))))) # label(fact_5032_LIMSEQ__realpow__zero) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2717 (all X_1 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(sin,hAPP_real_real(uminus_uminus_real,X_1))),hAPP_real_real(sin,X_1))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(cos,hAPP_real_real(uminus_uminus_real,X_1))),hAPP_real_real(cos,X_1))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_real) # label(fact_3741_sin__cos__minus) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2718 (all B_5 all A_5 all A_3 all B_2 (zero_zero_nat != hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2) -> (A_3 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_5),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)) -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_5),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)) = B_2 -> one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_5),B_5))))) # label(fact_4364_gcd__coprime__nat) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2719 (all X_1 all Y_1 hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(re,X_1)),hAPP_complex_real(re,Y_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(im,X_1)),hAPP_complex_real(im,Y_1)))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(re,X_1)),hAPP_complex_real(im,Y_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(im,X_1)),hAPP_complex_real(re,Y_1)))) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_1),Y_1)) # label(fact_4135_complex__mult__def) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2720 (all A_91 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_91),one_one_nat) = A_91) # label(fact_2101_mult_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2721 (all A_3 one_one_real = hAPP_complex_real(norm_norm_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(cos,A_3)),hAPP_real_real(sin,A_3)))) # label(fact_3965_cmod__unit__one) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2722 (all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) <-> zero_zero_nat != N)) # label(fact_1741_neq0__conv) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2723 (all Ma all N all A (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A)) -> (N = Ma <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A),Ma)))) # label(fact_873_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2724 (all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),one_one_real)) -> hAPP_real_real(cos,hAPP_real_real(arccos,Y_1)) = Y_1))) # label(fact_3606_cos__arccos) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2725 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B)))))) # label(fact_4724_zero__le__Fract__iff) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2726 (all X_1 all Y_1 (X_1 = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> zero_zero_real = Y_1)) # label(fact_3348_real__sqrt__sum__squares__eq__cancel) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2727 (all A all B all C all D (hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C),D) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C),D)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),B))))) # label(fact_2094_diff__eq__diff__less__eq) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2728 (all A_112 all B_91 all C_51 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_112),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_91),C_51)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_112),B_91)),C_51)) # label(fact_1903_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2729 (all V_7 all B_99 all C_55 hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_7)),B_99)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_7)),C_55)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_7)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,B_99),C_55))) # label(fact_1558_right__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2730 (all Ma (zero_zero_nat != Ma -> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,dvd_dvd_nat),Ma)))))) # label(fact_4753_finite__divisors__nat) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2731 (all V all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_nat_nat(suc,N))) <-> (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))),N))))) # label(fact_4782_less__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 5.56/5.65 2732 (all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),M)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(one_one_int),hAPP_int_int(number_number_of_int,min)),M)))) # label(fact_1504_one__not__neg__one__mod__m) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2733 (all A all B (zero_zero_real = B & A = one_one_real <-> one_one_complex = hAPP_real_complex(hAPP_r265291036omplex(complex,A),B))) # label(fact_3959_Complex__eq__1) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2734 (all X_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))) -> zero_zero_nat = X_1 | hAPP_nat_nat(suc,zero_zero_nat) = X_1)) # label(fact_3051_nat__lt__two__imp__zero__or__one) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2735 (all N_32 all M_18 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),M_18)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),N_32)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,M_18),N_32)))))) # label(fact_1367_less__1__mult) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2736 (all A_121 all C_56 all B_101 all D_9 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_121),B_101)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C_56),D_9)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_121),C_56)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_101),D_9))) # label(fact_1525_add__diff__add) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2737 (all Y_15 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,zero_zero_real),Y_15) = zero_zero_real) # label(fact_2472_mult__left_Ozero) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2738 (all A_50 all C_15 all B_42 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,C_15),B_42)) -> hAPP_int_int(div_mod_int(hAPP_int_int(div_mod_int(A_50),B_42)),C_15) = hAPP_int_int(div_mod_int(A_50),C_15))) # label(fact_2388_mod__mod__cancel) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2739 (all B_5 all A_5 all A_3 all B_2 (zero_zero_int != hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2) -> (hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_5),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)) = A_3 -> (hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_5),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)) = B_2 -> one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_5),B_5))))) # label(fact_4967_gcd__coprime__int) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2740 (all C_38 all A_95 all B_78 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_95),B_78)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_38),A_95)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_38),B_78))))) # label(fact_2058_add__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2741 (all W all Z1 all Z2 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z2)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z1),Z2))) # label(fact_1534_zdiff__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2742 (all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),hAPP_int_int(bit1,K_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),K_1)))) # label(fact_936_rel__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2743 (all A all N (hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A),N) = zero_z891286103de_int <-> A = zero_z891286103de_int & N != zero_zero_nat)) # label(fact_551_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2744 (all P_1 all A all B all N hAPP_P1860904029l_real(produc595218619l_real(hAPP_f2072293513l_real(hAPP_f1763467369l_real(cOMBS_163651580l_real,hAPP_f1663701695l_real(hAPP_f774948957l_real(cOMBB_3698607l_real,cOMBS_902912708l_real),hAPP_f324002831l_real(hAPP_f677014621l_real(cOMBS_107495110l_real,hAPP_f1161935955l_real(hAPP_f328572565l_real(cOMBB_520243667l_real,cOMBS_794971624l_real),hAPP_f517738447l_real(hAPP_f1879982819l_real(cOMBB_731595089l_real,hAPP_f1076489843l_real(cOMBB_415306361l_real,if_Pro313124157l_real)),hAPP_f1884005689l_bool(hAPP_f172451459l_bool(cOMBB_421479067l_real,hAPP_f940061907l_bool(cOMBB_1795223523l_real,P_1)),hAPP_f320596107l_real(hAPP_f850851693l_real(cOMBS_193881978l_real,hAPP_f1119546819l_real(hAPP_f555557763l_real(cOMBB_422423973l_real,cOMBB_2107848501l_real),produc865579683l_real)),hAPP_r337325687l_real(hAPP_f1644353557l_real(cOMBC_1508019322l_real,hAPP_f1644353557l_real(hAPP_f1874145443l_real(cOMBB_1768400739l_real,cOMBC_real_real_real),hAPP_f1732848529l_real(hAPP_f1310479593l_real(cOMBB_2092576745l_real,hAPP_f1960414057l_real(cOMBB_882763271l_real,inverse_divide_real)),plus_plus_real))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))),hAPP_f1591894897l_real(hAPP_f644105375l_real(cOMBC_467724138l_real,hAPP_f1108539711l_real(hAPP_f1109256153l_real(cOMBB_44084907l_real,cOMBS_511527878l_real),hAPP_f823702159l_real(hAPP_f1040184293l_real(cOMBB_557258983l_real,hAPP_f2114902373l_real(cOMBB_237479429l_real,produc865579683l_real)),hAPP_r337325687l_real(hAPP_f1644353557l_real(cOMBC_1508019322l_real,hAPP_f1644353557l_real(hAPP_f1874145443l_real(cOMBB_1768400739l_real,cOMBC_real_real_real),hAPP_f1732848529l_real(hAPP_f1310479593l_real(cOMBB_2092576745l_real,hAPP_f1960414057l_real(cOMBB_882763271l_real,inverse_divide_real)),plus_plus_real))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),cOMBI_real)))),hAPP_f320596107l_real(hAPP_f850851693l_real(cOMBS_193881978l_real,hAPP_f1119546819l_real(hAPP_f555557763l_real(cOMBB_422423973l_real,cOMBB_2107848501l_real),produc865579683l_real)),hAPP_r337325687l_real(hAPP_f1644353557l_real(cOMBC_1508019322l_real,hAPP_f1644353557l_real(hAPP_f1874145443l_real(cOMBB_1768400739l_real,cOMBC_real_real_real),hAPP_f1732848529l_real(hAPP_f1310479593l_real(cOMBB_2092576745l_real,hAPP_f1960414057l_real(cOMBB_882763271l_real,inverse_divide_real)),plus_plus_real))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N)) = hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),hAPP_nat_nat(suc,N))) # label(fact_4851_Bolzano__bisect_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2745 (all X_22 all Y_17 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_22),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_17),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_22)),Y_17)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_22),Y_17)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_1619_power2__diff) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2746 (all A_59 all B_51 all C_22 hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_59),hAPP_C498520661umeral(div_mo1740067990umeral(B_51),C_22))),C_22) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_59),B_51)),C_22)) # label(fact_2364_zmod__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2747 (all N_55 all A_205 all B_156 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_205),B_156)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_205)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_205),N_55)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_156),N_55)))))) # label(fact_656_power__mono) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2748 (all D all Sa exists Z (is_int(Z) & (all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z),X)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Sa))) <-> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Sa)))))))) # label(fact_2261_pinf_I10_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2749 (all M_8 all N_22 hAPP_nat_rat(semiri151668891at_rat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_8),N_22)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(semiri151668891at_rat,M_8)),hAPP_nat_rat(semiri151668891at_rat,N_22))) # label(fact_2956_of__nat__add) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2750 (all B_1_1 all B_2_1 is_bool(hAPP_f1469637905l_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__fun_Itc__prod_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__f) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2751 (all A_18 all B_15 A_18 = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_18),B_15)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_18),B_15)),B_15))) # label(fact_2669_semiring__div__class_Omod__div__equality_H) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2752 (all A_11 all N_18 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_11),hAPP_nat_nat(suc,N_18)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_11),N_18)),A_11)) # label(fact_3033_power__Suc2) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2753 (all N_1 all X_1 (is_int(N_1) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(real_int,N_1)),one_one_real)),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_int_real(real_int,N_1))) -> N_1 = archim856651990g_real(X_1))))) # label(fact_4469_ceiling__eq3) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2754 (all Lx_3 all Ly_1 all Rx_3 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_3),Rx_3)),Ly_1) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_3),Ly_1)),Rx_3)) # label(fact_158_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2755 (all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K_1)),min)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),min)))) # label(fact_935_rel__simps_I13_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2756 (all N_1 all K_2 all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),M)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N_1)))))) # label(fact_4320_gcd__greatest__nat) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2757 (all P all Q all R hAPP_int_int(hAPP_int_fun_int_int(P,R),Q) = hAPP_int_int(hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Int__Oint_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2758 (all N_1 all A_3 hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),A_3)) = hAPP_complex_real(re,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,cis(A_3)),N_1))) # label(fact_4144_cos__n__Re__cis__pow__n) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2759 (all P_1 all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) -> (hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,minus_minus_real),hAPP_f73570213t_real(hAPP_f918606171t_real(cOMBB_64450052al_nat,produc1935615926l_real),bolzano_bisect(P_1,A,B)))),hAPP_f73570213t_real(hAPP_f918606171t_real(cOMBB_64450052al_nat,produc556554744l_real),bolzano_bisect(P_1,A,B))),zero_zero_real,sequentially)) -> (exists L_1 (hBOOL(tendsto_nat_real(hAPP_f73570213t_real(hAPP_f918606171t_real(cOMBB_64450052al_nat,produc556554744l_real),bolzano_bisect(P_1,A,B)),L_1,sequentially)) & (all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,L_1),hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_2))))) & hBOOL(tendsto_nat_real(hAPP_f73570213t_real(hAPP_f918606171t_real(cOMBB_64450052al_nat,produc1935615926l_real),bolzano_bisect(P_1,A,B)),L_1,sequentially)) & (all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_2))),L_1)))))))))) # label(fact_5058_Bolzano__nest__unique) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2760 (all C_43 all A_100 all B_83 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_100),B_83)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_43),A_100)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_43),B_83))))) # label(fact_2012_add__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2761 (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2)) -> (zero_zero_real = hAPP_real_real(ln,X_2) <-> one_one_real = X_2))) # label(fact_3442_ln__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2762 (all X_21 all N_31 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31)) -> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_21),N_31) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_21),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_31),one_one_nat))),X_21))) # label(fact_1735_realpow__minus__mult) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2763 (all Ma (one_one_int = hAPP_int_int(abs_abs_int,Ma) <-> hAPP_int_int(legacy_zgcd(zero_zero_int),Ma) = one_one_int)) # label(fact_4518_zgcd__0__1__iff) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2764 (all B all A (B = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B),A) <-> zero_zero_nat = A)) # label(fact_637_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2765 (all X_1 all M all Y_1 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(div_mod_int(X_1),M)),Y_1)),M) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),Y_1)),M)) # label(fact_2399_zdiff__zmod__left) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2766 (all X_2 all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,X_2)),hAPP_int_int(number_number_of_int,Y_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),Y_2)))) # label(fact_93_le__number__of) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2767 (all P all Q all R hAPP_f1736103445l_real(hAPP_r224952523l_real(P,R),hAPP_r2019696015l_real(Q,R)) = hAPP_r2019696015l_real(hAPP_f2072293513l_real(hAPP_f1763467369l_real(cOMBS_163651580l_real,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__p) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2768 (all A_104 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A_104) = A_104) # label(fact_1993_add__0__left) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2769 (all A_3 all B_2 all C_1 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(div_mod_int(B_2),C_1))),C_1) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)),C_1)) # label(fact_2397_zmod__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2770 (all V (zero_zero_nat = hAPP_int_nat(number_number_of_nat,V) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V),pls)))) # label(fact_1001_eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2771 (all A_3 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(suc,zero_zero_nat)),A_3))) # label(fact_4021_coprime__Suc0_H) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2772 (all Ma all N (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(semiri132038758t_real,Ma)),hAPP_nat_real(semiri132038758t_real,N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)))) # label(fact_2953_of__nat__le__iff) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2773 (all N_1 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N_1)),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,N_1)),one_one_real))) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),one_one_nat) = natceiling(X_1)))) # label(fact_3724_natceiling__eq) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2774 (all X_1 all Y_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(fequal_nat,X_1),Y_1)) | X_1 != Y_1)) # label(help_fequal_2_1_fequal_000tc__Nat__Onat_T) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2775 (all Y_1 all X_1 all U_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),U_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,U_1),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,U_1),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),U_1)))))))) # label(fact_3351_lemma__sqrt__hcomplex__capprox) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2776 (all C all A all B (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A),B)) <-> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C),A)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C),B))))) # label(fact_2066_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2777 (all A_139 all B_116 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_139),B_116))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_139)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_116))))) # label(fact_1298_zero__less__mult__pos) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2778 (all Z1 all Z2 all W hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z1),Z2)),W) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),W)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z2),W))) # label(fact_1535_zdiff__zmult__distrib) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2779 (all A all D all N hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,N)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),D)))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,hAPP_f1139079189at_int(hAPP_f1431025877at_int(cOMBB_int_int_nat,hAPP_int_fun_int_int(plus_plus_int,A)),hAPP_int_fun_nat_int(hAPP_f1673103573at_int(cOMBC_nat_int_int,hAPP_f1935805932nt_int(hAPP_f125788821nt_int(cOMBB_859312247nt_nat,times_times_int),semiri1621563631at_int)),D))),hAPP_n1699378549t_bool(ord_lessThan_nat,N)))) # label(fact_4946_arith__series__int) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2780 (all A_189 all N_40 all N_39 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_40),N_39)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,one_on1684967323de_int),A_189)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_189),N_40)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_189),N_39)))))) # label(fact_870_power__increasing) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2781 (all B adjust(B) = produc1518849193nt_int(hAPP_f465821005nt_int(hAPP_f1081447804nt_int(cOMBS_1470252563nt_int,hAPP_f617310012nt_int(hAPP_f197974549nt_int(cOMBB_638531587nt_int,cOMBS_1891347093nt_int),hAPP_f55007289nt_int(hAPP_f747475781nt_int(cOMBB_222539774nt_int,hAPP_f1311564564nt_int(cOMBS_1640193733nt_int,hAPP_f251264923nt_int(hAPP_f279307923nt_int(cOMBB_662120866nt_int,if_Pro1731782967nt_int),hAPP_f429935748t_bool(hAPP_f1512942609t_bool(cOMBB_int_bool_int,hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int)),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,minus_minus_int),B))))),hAPP_f1574055885nt_int(hAPP_f960630521nt_int(cOMBC_1805044090nt_int,hAPP_f142232355nt_int(hAPP_f1377453843nt_int(cOMBB_320287386nt_int,cOMBB_1846624359nt_int),hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,plus_plus_int),hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_int)))),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,minus_minus_int),B))))),hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_4840_adjust__def) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2782 (all A_3 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(inv(P_3),A_3))),one_one_int),P_3)))))) # label(fact_2147_inv__is__inv) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2783 (all X_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),hAPP_int_int(bit1,pls))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,X_2)),one_one_real)))) # label(fact_285_less__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2784 (all B_126 all A_150 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_150)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_126),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_150),B_126)),zero_zero_int))))) # label(fact_1233_mult__nonneg__nonpos) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2785 (all A all C all B (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),C))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B),A)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C),zero_zero_rat)) | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A),B)))) # label(fact_1328_mult__less__cancel__right__disj) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2786 (all M_9 all N_23 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_9)),hAPP_nat_int(semiri1621563631at_int,N_23))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_23)))) # label(fact_2948_of__nat__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2787 (all A all X_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),archim856651990g_real(X_2))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(real_int,A)),one_one_real)),X_2)))) # label(fact_4471_le__ceiling__eq) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2788 (all B_2 all N_1 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),hAPP_real_real(log(B_2),X_1)) = hAPP_real_real(log(B_2),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1)))) # label(fact_3871_log__nat__power) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2789 (all A_3 all B_2 hAPP_int_int(uminus_uminus_int,hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,A_3)),B_2)) = hAPP_int_int(div_mod_int(A_3),hAPP_int_int(uminus_uminus_int,B_2))) # label(fact_3523_zmod__zminus2) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2790 (all Lx_4 all Ly_2 all Rx_4 all Ry_2 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_4),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Ly_2),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_4),Ry_2))) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_4),Ly_2)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_4),Ry_2))) # label(fact_155_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2791 (all I all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,I),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,I),N_1))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),I)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))))) # label(fact_947_power__dvd__imp__le) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2792 (all B_1_1 all B_2_1 is_bool(hAPP_nat_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__Nat__Onat_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2793 (all C_83 all D_16 all A_169 all B_144 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_169),B_144)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,C_83),D_16)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_169),C_83)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_144),D_16)))))) # label(fact_1109_mult__dvd__mono) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2794 (all P all Q all R hAPP_int_fun_int_int(P,hAPP_int_int(Q,R)) = hAPP_int_fun_int_int(hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J) # label(axiom) # label(non_clause). [assumption]. 5.56/5.66 2795 (all Y_2 hAPP_real_real(arctan,Y_2) = the_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,ord_less_real),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),tan)),Y_2))))) # label(fact_4881_arctan__def) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2796 (all I_1 all J hAPP_n1699378549t_bool(ord_at238088361st_nat(hAPP_nat_nat(suc,I_1)),hAPP_nat_nat(suc,J)) = image_nat_nat(suc,hAPP_n1699378549t_bool(ord_at238088361st_nat(I_1),J))) # label(fact_5071_image__Suc__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2797 (all I all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(nat,I)),N_1) = hAPP_nat_nat(nat_aux(I),N_1)) # label(fact_2778_nat__aux__def) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2798 (all B_113 all A_136 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_136)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_113)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_136),B_113)))))) # label(fact_1316_mult__pos__pos) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2799 (all X_2 quotient_of(X_2) = the_Pr2103884470nt_int(hAPP_f237669397t_bool(hAPP_f461489489t_bool(cOMBS_400904860l_bool,hAPP_f365317245l_bool(hAPP_f1961001761l_bool(cOMBB_1550063917nt_int,fconj),hAPP_f1084724746t_bool(hAPP_f374502857t_bool(cOMBB_569260360nt_int,hAPP_r2115560837t_bool(fequal_rat,X_2)),hAPP_f1065357839nt_rat(hAPP_f1120137083nt_rat(cOMBS_549334880nt_rat,hAPP_f1547007718nt_rat(hAPP_f382317021nt_rat(cOMBB_170828966nt_int,fract),product_fst_int_int)),product_snd_int_int)))),hAPP_f237669397t_bool(hAPP_f461489489t_bool(cOMBS_400904860l_bool,hAPP_f365317245l_bool(hAPP_f1961001761l_bool(cOMBB_1550063917nt_int,fconj),hAPP_f892584630t_bool(hAPP_f492612297t_bool(cOMBB_798864284nt_int,hAPP_i1948725293t_bool(ord_less_int,zero_zero_int)),product_snd_int_int))),hAPP_i119638232t_bool(hAPP_f2018027565t_bool(cOMBC_33652895t_bool,hAPP_f291087797t_bool(hAPP_f214467655t_bool(cOMBB_182850971nt_int,fequal_int),hAPP_f521271395nt_int(hAPP_f1914742907nt_int(cOMBS_2019229620nt_int,hAPP_f1855025978nt_int(hAPP_f1038672605nt_int(cOMBB_1506897658nt_int,gcd_gcd_int),product_fst_int_int)),product_snd_int_int))),one_one_int))))) # label(fact_5022_quotient__of__def) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2800 (all R_3 all X_1 all Y_1 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,of_real_complex(R_3)),hAPP_real_complex(hAPP_r265291036omplex(complex,X_1),Y_1)) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,R_3),X_1)),Y_1)) # label(fact_3991_complex__of__real__add__Complex) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2801 (all A all B (hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B) = zero_zero_rat <-> A = B)) # label(fact_2093_right__minus__eq) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2802 (all A big_co1548731110nt_int(cOMBI_int,hAPP_i1948725293t_bool(d22set,A)) = hAPP_int_int(zfact,A)) # label(fact_4809_d22set__prod__zfact) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2803 (all Y_1 all X_1 all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> (one_one_real != A_3 -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1)) -> hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(log(A_3),X_1)),hAPP_real_real(log(A_3),Y_1)) = hAPP_real_real(log(A_3),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_1),Y_1))))))) # label(fact_3873_log__divide) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2804 (all A_27 all B_24 (is_int(B_24) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_27),B_24)) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_27),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_24),A_27)) = B_24))) # label(fact_2591_dvd__mult__div__cancel) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2805 (all A_109 all B_88 all C_48 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_109),B_88) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_109),C_48) -> C_48 = B_88)) # label(fact_1926_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2806 (all X_1 all M hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),hAPP_int_int(div_mod_int(X_1),M)),M))) # label(fact_2402_mod__mod__is__mod) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2807 (all Q all P (hBOOL(fdisj(P,Q)) | -hBOOL(P))) # label(help_fdisj_1_1_U) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2808 (all P_1 P_1 = collect_int(P_1)) # label(fact_206_Collect__def) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2809 (all Real1_2 all Real2_2 all Real1_1 all Real2_1 (hAPP_real_complex(hAPP_r265291036omplex(complex,Real1_2),Real2_2) = hAPP_real_complex(hAPP_r265291036omplex(complex,Real1_1),Real2_1) <-> Real2_2 = Real2_1 & Real1_1 = Real1_2)) # label(fact_3955_complex_Oinject) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2810 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sqrt,X_1))))) # label(fact_3323_real__sqrt__ge__zero) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2811 (all A_239 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_239),A_239) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_239),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_39_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2812 (all C_66 all D_13 all A_132 all B_111 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_132),B_111)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_66),D_13)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_111)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_66)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_132),C_66)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_111),D_13)))))))) # label(fact_1411_mult__strict__mono) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2813 (all Xa_4 all X_18 all Y_14 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa_4),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_18),Y_14)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa_4),X_18)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa_4),Y_14))) # label(fact_2480_mult__right_Oadd) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2814 (all L all U hAPP_i1948725293t_bool(ord_at641636577an_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,L),one_one_int)),U) = hAPP_i1948725293t_bool(ord_gr1297742076an_int(L),U)) # label(fact_4920_atLeastPlusOneLessThan__greaterThanLessThan__int) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2815 (all A_3 all B_2 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),B_2))) # label(fact_1614_zspecial__product) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2816 (all A_87 all B_76 (is_int(A_87) -> A_87 = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_87),B_76)),B_76))) # label(fact_2133_add__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2817 (all Y_3 all X_3 (hBOOL(vanishes(X_3)) -> (hBOOL(vanishes(Y_3)) -> hBOOL(vanishes(hAPP_f2062659861at_rat(hAPP_f1131305186at_rat(cOMBS_nat_rat_rat,hAPP_f1830834240at_rat(hAPP_f1073644437at_rat(cOMBB_1926947at_nat,plus_plus_rat),X_3)),Y_3)))))) # label(fact_5130_vanishes__add) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2818 (all A_49 all B_41 zero_z891286103de_int = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_49),B_41)),B_41)) # label(fact_2410_mod__mult__self2__is__0) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2819 (all Z_12 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_12) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Z_12),Z_12)) # label(fact_488_semiring__mult__2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2820 (all A_68 all C_28 all B_60 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_68),C_28)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_60),C_28)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(div_mod_int(A_68),B_60)),C_28)) # label(fact_2330_mod__mult__mult2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2821 (all V_14 all W_12 hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_14),W_12)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_14)),hAPP_int_real(number267125858f_real,W_12))) # label(fact_430_number__of__mult) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2822 (all M all V_2 hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,M)))),hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),M)),hAPP_int_nat(number_number_of_nat,V_2))) # label(fact_3209_Suc__mod__eq__add3__mod__number__of) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2823 (all M all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,N_1)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),N_1))) # label(fact_3208_div__Suc__eq__div__add3) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2824 (all L_2 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,min),hAPP_int_int(bit1,L_2)) = hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,min),L_2))) # label(fact_1589_diff__bin__simps_I6_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2825 (all X_2 all Y_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(semiri1621563631at_int,X_2)),hAPP_nat_int(semiri1621563631at_int,Y_2))))) # label(fact_3046_Nat__Transfer_Otransfer__int__nat__relations_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2826 (all M all N_1 all K_2 (hAPP_int_int(legacy_zgcd(N_1),K_2) = one_one_int -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N_1))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M)))))) # label(fact_4526_zrelprime__zdvd__zmult__aux) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2827 (all R_3 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,R_3),hAPP_int_real(real_int,archim856651990g_real(R_3))))) # label(fact_4401_real__of__int__ceiling__ge) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2828 (all A_187 all N_37 all N_36 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_37),N_36)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_187)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_187),N_37)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_187),N_36)))))) # label(fact_895_power__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2829 (all N all Ma (one_one_nat = Ma & one_one_nat = N <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),Ma) = one_one_nat)) # label(fact_1615_nat__mult__eq__one) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2830 (all A_202 all N_52 all N_51 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_52),N_51)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_202)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_202),one_one_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_202),N_51)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_202),N_52))))))) # label(fact_680_power__decreasing) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2831 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),M)))))) # label(fact_3130_n__less__n__mult__m) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2832 (all X_37 all Y_29 (hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_29),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_37),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_37)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_29)) -> Y_29 = X_37)))) # label(fact_791_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2833 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (hAPP_real_real(ln,X_1) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),one_one_real) -> one_one_real = X_1))) # label(fact_3431_ln__eq__minus__one) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2834 (all B_115 all A_138 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_138)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_115),zero_zero_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_115),A_138)),zero_zero_rat))))) # label(fact_1307_mult__pos__neg2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2835 (all X_1 all Y_1 all M (hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_1),Y_1)),M) = one_one_nat -> one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),M))) # label(fact_4370_invertible__coprime__nat) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2836 (all P all Q all R hAPP_nat_nat(P,hAPP_nat_nat(Q,R)) = hAPP_nat_nat(hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__Nat__Onat_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2837 (all N_12 all R_6 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),R_6)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,R_6),one_one_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,R_6),hAPP_nat_nat(suc,N_12))),R_6))))) # label(fact_3160_realpow__Suc__le__self) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2838 (all X_1 (is_int(X_1) -> (-hBOOL(hAPP_int_bool(even_odd_even_int,X_1)) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_int) = X_1))) # label(fact_2806_two__times__odd__div__two__plus__one) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2839 (all D_3 all C_1 all A_3 all B_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M)) -> (C_1 = B_2 -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C_1),D_3),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),D_3),M)))))) # label(fact_1004_zcong__eq__trans) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2840 (all P_1 collect_nat(P_1) = image_int_nat(nat,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_f1448486952t_bool(hAPP_f696844121t_bool(cOMBB_nat_bool_int,P_1),nat))))) # label(fact_4748_Nat__Transfer_Otransfer__nat__int__set__functions_I5_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2841 (all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2)) -> (exists K (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K)) & J_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K))))) # label(fact_2244_less__imp__add__positive) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2842 (all F all L (hBOOL(tendsto_nat_real(F,L,sequentially)) -> hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,abs_abs_real),F),hAPP_real_real(abs_abs_real,L),sequentially)))) # label(fact_5040_LIMSEQ__imp__rabs) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2843 (all Z_1 all W hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_1),W) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),hAPP_int_int(uminus_uminus_int,W))) # label(fact_3535_diff__int__def) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2844 (all A_102 all C_45 all B_85 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_102),C_45)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_85),C_45))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_102),B_85)))) # label(fact_2003_add__le__imp__le__right) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2845 (all X_2 (hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(suc,X_2))) <-> -hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2)))) # label(fact_3758_even__Suc) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2846 (all X_1 all N_5 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_5)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),hAPP_real_real(hAPP_n546711566l_real(root,N_5),X_1)))))))) # label(fact_3936_real__root__increasing) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2847 (all P all Q all R hAPP_n682674106l_real(hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,P),Q),R) = hAPP_b646336293l_real(P,hAPP_nat_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__RealDef__Oreal_Mtc__fun_I) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2848 (all X_2 all Y_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_2)),Y_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_2),Y_2)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_10_power2__sum) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2849 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> one_one_real = hAPP_real_real(sgn_sgn_real,X_1))) # label(fact_3570_real__sgn__pos) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2850 (all A_215 all M_24 all N_64 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_24),N_64)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_215),M_24)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_215),N_64))))) # label(fact_556_le__imp__power__dvd) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2851 (all N hAPP_n1699378549t_bool(ord_atMost_nat,N) = image_int_nat(nat,hAPP_i1948725293t_bool(ord_at875362053st_int(zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N)))) # label(fact_5182_SetInterval_Otransfer__nat__int__set__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2852 (all X_1 hAPP_real_real(sqrt,hAPP_real_real(inverse_inverse_real,X_1)) = hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,X_1))) # label(fact_3828_real__sqrt__inverse) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2853 (all M all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),M))))) # label(fact_4192_prime__dvd__square) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2854 (all A_28 (A_28 != zero_z126310315umeral -> one_on1645066479umeral = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_28),A_28))) # label(fact_2587_div__self) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2855 (all X_1 hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_complex_real(re,X_1)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(im,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_complex_real(re,hAPP_complex_complex(invers1449016382omplex,X_1))) # label(fact_4138_complex__Re__inverse) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2856 (all K_2 all M all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),hAPP_nat_nat(div_mod_nat(M),N_1)) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_1))) # label(fact_2509_mod__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2857 (all A_3 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(inv(P_3),A_3))))))) # label(fact_2131_inv__g__1) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2858 (all B_95 all C_53 all A_116 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_116)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_95),C_53)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_95),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_116),C_53)))))) # label(fact_1873_add__strict__increasing2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2859 (all Y_1 all X_1 all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,Y_1),A_3))))))) # label(fact_4404_powr__less__mono2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2860 (all A all C all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),C)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),C))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),A)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C),zero_zero_int)))) # label(fact_1326_mult__less__cancel__right__disj) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2861 (all B all A (is_int(B) & is_int(A) -> (A = zero_zero_int <-> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B),A) = B))) # label(fact_636_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2862 (all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arcsin,Y_1))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arcsin,Y_1)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3507_arcsin__bounded) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2863 (all A_174 all B_148 all C_86 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_174),C_86)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_148),C_86)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_174),B_148)),C_86)) # label(fact_1075_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2864 (all N_63 all M_23 all X_42 all Y_34 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_42),Y_34)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_63),M_23)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_42),N_63)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_34),M_23)))))) # label(fact_563_dvd__power__le) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2865 (all B_1_1 all B_2_1 is_bool(hAPP_P1333854989l_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc__HOL__Obo) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2866 (all C all B all A (is_int(A) & is_int(B) & is_int(C) -> (A != zero_zero_int -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),B)) -> (hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B),A) = C <-> B = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),A)))))) # label(fact_2665_dvd__div__eq__mult) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2867 (all B all A all C (B = C <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C),A))) # label(fact_1933_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2868 (all K_2 hAPP_int_rat(ring_1_of_int_rat,K_2) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,K_2),one_one_int)) # label(fact_4678_Fract__of__int__eq) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2869 (all K_2 all N_1 all M hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,M)),N_1) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_1)),M))),N_1)) # label(fact_3143_mod__mult__self4) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2870 (all A_42 all B_35 all B_34 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_42),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_35),B_34)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_42),B_35)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_42),B_34))) # label(fact_2478_mult_Oadd__right) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2871 (all A_89 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),A_89) = A_89) # label(fact_2115_mult__1) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2872 (all A_3 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,X_1)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3)),A_3)))))) # label(fact_4376_ln__powr__bound) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2873 (all X_2 hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,X_2))))) # label(fact_4834_summable__exp) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2874 (all L all U hAPP_i1948725293t_bool(ord_at875362053st_int(L),U) = hAPP_i1948725293t_bool(ord_at641636577an_int(L),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,U),one_one_int))) # label(fact_5079_atLeastLessThanPlusOne__atLeastAtMost__int) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2875 (all K_2 all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_2),K_2))))) # label(fact_1789_mult__le__mono1) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2876 (all C_68 all A_143 all B_120 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_143),B_120)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_68)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_143),C_68)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_120),C_68)))))) # label(fact_1276_mult__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2877 (all K_2 hAPP_int_int(pred,hAPP_int_int(bit0,K_2)) = hAPP_int_int(bit1,hAPP_int_int(pred,K_2))) # label(fact_4040_pred__Bit0) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2878 (all K_1 all L (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K_1)),hAPP_int_int(bit0,L))))) # label(fact_268_rel__simps_I16_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2879 (all A_170 all B_145 all K_7 (hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_145),K_7) = A_170 -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,B_145),A_170)))) # label(fact_1102_dvdI) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2880 (all X_7 all Q_5 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_7),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_7),Q_5)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_7),hAPP_nat_nat(suc,Q_5))) # label(fact_3025_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2881 (all R_3 all S_1 hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(number267125858f_real,R_3)),hAPP_int_real(number267125858f_real,S_1)) = ratreal(hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,hAPP_int_rat(number_number_of_rat,R_3)),hAPP_int_rat(number_number_of_rat,S_1)))) # label(fact_4667_Ratreal__number__of__quotient2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2882 (all P all Q all R hAPP_i61245889nt_int(P,hAPP_P1175774780nt_int(Q,R)) = hAPP_P1668131738nt_int(hAPP_f167782601nt_int(hAPP_f400538607nt_int(cOMBB_866192175nt_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__I) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2883 (all A_1 (hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1)) -> (hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_1)) <-> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_int_nat(nat,A_1)))))) # label(fact_4618_transfer__int__nat__set__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2884 (all N_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),hAPP_int_int(fact_fact_int,N_1))))) # label(fact_4281_fact__ge__one__int) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2885 (all A_67 all C_27 all B_59 hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_67),C_27)),B_59)),C_27) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_67),B_59)),C_27)) # label(fact_2337_zmod__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2886 (all X_22 all Y_17 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_22),Y_17)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_22),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_17),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_22)),Y_17))) # label(fact_1618_power2__diff) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2887 (all A_71 all B_63 all C_31 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_71),B_63)),C_31) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(div_mod_int(A_71),C_31)),B_63)),C_31)) # label(fact_2318_mod__mult__left__eq) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2888 (all M all N_1 hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N_1) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M),N_1))) # label(fact_4971_gcd__add2__int) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2889 (all K_1 all F (hBOOL(summable_real(F)) -> ((all D_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),D_2)))),hAPP_nat_real(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),D_2)),one_one_nat))))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),K_1))),hAPP_f352196356l_real(suminf_real,F)))))) # label(fact_4926_sumr__pos__lt__pair) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2890 (all X_2 (is_int(X_2) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven)) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_2),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = X_2))) # label(fact_3238_even__div__2__prop2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2891 (all Ma all N all A (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A)) -> (N = Ma <-> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),N) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),Ma)))) # label(fact_872_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2892 (all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_n522471361de_int(semiri1424489471de_int,N))))) # label(fact_3146_of__nat__0__less__iff) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2893 (all V_3 all K_2 all V_2 ((hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_3)),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_3)),K_2)) & (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> (hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_3))) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_3)),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),K_2)) & (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_3))) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_3)),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_2),V_3))),K_2))))) # label(fact_3283_nat__number__of__add__left) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2894 (all B_12 all A_15 all C_5 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_12),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_15),B_12))),hAPP_nat_nat(div_mod_nat(A_15),B_12))),C_5) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_15),C_5)) # label(fact_2680_div__mod__equality2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2895 (all B_138 all A_162 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_138)) & hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_162),zero_z126310315umeral)) | hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_162)) & hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_138),zero_z126310315umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_162),B_138)),zero_z126310315umeral)))) # label(fact_1176_split__mult__neg__le) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2896 (all X_10 all Y_7 ((all Z (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),Z)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Z),one_one_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Z),X_10)),Y_7))))) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_10),Y_7)))) # label(fact_2818_field__le__mult__one__interval) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2897 (all C_33 all D_5 all A_78 all B_67 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_78),B_67)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_33),D_5)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_78),C_33)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_67),D_5)))))) # label(fact_2212_add__less__le__mono) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2898 (all A_223 A_223 = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,pls)),A_223)) # label(fact_415_add__numeral__0) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2899 (all I all J_2 hAPP_int_int(legacy_zgcd(I),J_2) = hAPP_int_int(legacy_zgcd(I),hAPP_int_int(uminus_uminus_int,J_2))) # label(fact_4495_zgcd__zminus2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2900 (all A_171 all B_146 all C_84 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_171),B_146)),C_84)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_171),C_84)))) # label(fact_1095_dvd__mult__left) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2901 (all X_43 all Y_35 all Q_10 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_43),Y_35)),Q_10) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_43),Q_10)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_35),Q_10))) # label(fact_344_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2902 (all X_2 (X_2 = zero_z126310315umeral <-> X_2 = zero_z126310315umeral)) # label(fact_1899_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2903 (all P_5 collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),P_5))) = hAPP_i1948725293t_bool(sr,P_5)) # label(fact_2291_SR__def) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2904 (all A_107 A_107 = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_107),zero_z891286103de_int)) # label(fact_1968_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2905 (all Y_1 all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1)) -> hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_int_nat(nat,X_1)),hAPP_int_nat(nat,Y_1))))) # label(fact_5019_transfer__nat__int__gcd_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2906 (all A_192 all M_20 all N_43 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_192),M_20)),N_43) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_192),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_20),N_43))) # label(fact_837_power__mult) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2907 (all X_2 all Y_2 (is_int(Y_2) & is_int(X_2) -> (zero_zero_int = Y_2 & zero_zero_int = X_2 <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),zero_zero_int))))) # label(fact_848_sum__power2__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2908 (all Z_1 hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,Z_1),of_real_complex(hAPP_complex_real(norm_norm_complex,Z_1))) = hAPP_complex_complex(sgn_sgn_complex,Z_1)) # label(fact_3988_sgn__eq) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2909 (all M all K_2 all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),N_1))) # label(fact_4347_gcd__add__mult__nat) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2910 (all N_1 (-hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1)) -> hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),one_one_nat)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3789_lemma__not__even__div2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2911 (all X_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,natfloor(X_1))),one_one_real)))) # label(fact_3689_real__natfloor__add__one__gt) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2912 (all A_166 all B_141 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_166),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_141),A_166)))) # label(fact_1131_dvd__triv__right) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2913 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(ln,X_1))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X_1))))) # label(fact_3451_ln__ge__zero__imp__ge__one) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2914 (all C_66 all D_13 all A_132 all B_111 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_132),B_111)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,C_66),D_13)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_111)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_66)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_132),C_66)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_111),D_13)))))))) # label(fact_1410_mult__strict__mono) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2915 (all X_43 all Y_35 all Q_10 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_43),Y_35)),Q_10) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_43),Q_10)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_35),Q_10))) # label(fact_345_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2916 (all C_60 all A_126 all B_105 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_60),A_126)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_60),B_105))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_60)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_126),B_105))))) # label(fact_1447_mult__left__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2917 (all R_3 all Q_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,R_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),Q_2)) = A_3 -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_2),one_one_int)))))) # label(fact_903_self__quotient__aux2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2918 (all N_1 all I (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),I)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,I),N_1))))) # label(fact_3142_nat__one__le__power) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2919 (all M all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),J_2))))) # label(fact_1762_trans__less__add2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2920 (all A_58 all C_21 all B_50 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(div_mod_int(A_58),C_21)),B_50)),C_21) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_58),B_50)),C_21)) # label(fact_2366_zmod__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2921 (all Ma all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) <-> (exists K hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),K)) = N))) # label(fact_3056_less__iff__Suc__add) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2922 (all P_1 all N ((exists M_1 (hBOOL(hAPP_nat_bool(P_1,M_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N)))) <-> (exists X (hBOOL(hAPP_nat_bool(P_1,X)) & hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),hAPP_n1699378549t_bool(ord_at238088361st_nat(zero_zero_nat),N))))))) # label(fact_5068_ex__nat__less) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2923 (all Ma all N (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,Ma)),hAPP_int_real(number267125858f_real,N))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ma),hAPP_int_int(number_number_of_int,N))))) # label(fact_4441_number__of__less__real__of__int__iff2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2924 (all Ma all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(nat_case_nat(zero_zero_nat,cOMBI_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),N))) # label(fact_4879_diff__Suc) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2925 (all N_12 all R_6 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),R_6)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,R_6),one_one_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,R_6),hAPP_nat_nat(suc,N_12))),R_6))))) # label(fact_3156_realpow__Suc__le__self) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2926 (all X_1 all Y_1 hAPP_complex_real(im,hAPP_real_complex(hAPP_r265291036omplex(complex,X_1),Y_1)) = Y_1) # label(fact_4091_Im) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2927 (all C_79 all D_15 all A_160 all B_136 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_160),B_136)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_79),D_15)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_136)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_79)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_160),C_79)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_136),D_15)))))))) # label(fact_1183_mult__mono) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2928 (all Q_2 all B_2 all R_3 all C_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),R_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_3),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(div_mod_int(Q_2),C_1))),R_3)),zero_zero_int)))))) # label(fact_2466_zmult2__lemma__aux2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2929 (all M_19 all N_38 all A_188 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_188)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_188),M_19)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_188),N_38))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_19),N_38))))) # label(fact_886_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2930 (all C_69 all A_144 all B_121 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_144),B_121)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_69)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_69),A_144)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_69),B_121)))))) # label(fact_1270_mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2931 (all X_24 all Y_18 all A_120 all B_100 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_24),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_18),B_100))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_24),A_120)),B_100)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_24),Y_18)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_120),B_100))) # label(fact_1540_mult__diff__mult) # label(axiom) # label(non_clause). [assumption]. 5.65/5.66 2932 (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X_2)),one_one_real)) -> hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,one_one_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_2)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))) = hAPP_real_real(arctan,X_2))) # label(fact_4816_arctan__series) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2933 (exists K_4 ((all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(im,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X)),K_4)))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),K_4)))) # label(fact_4106_Im_Opos__bounded) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2934 (all X_1 all Y_1 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X_1)),hAPP_real_real(cos,Y_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_1)),hAPP_real_real(sin,Y_1))) = hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_1))) # label(fact_3664_sin__add) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2935 (all N_19 zero_z126310315umeral = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,zero_z126310315umeral),hAPP_nat_nat(suc,N_19))) # label(fact_2996_power__0__Suc) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2936 (all B all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2)) -> hBOOL(hAPP_real_bool(deriv_real(log(B),X_2),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(ln,B)),X_2)))))) # label(fact_4790_DERIV__log) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2937 (all B_71 all A_82 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_82)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_71)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_82),B_71)))))) # label(fact_2173_add__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2938 (all L all U hAPP_f22106695ol_nat(finite_card_nat,hAPP_n1699378549t_bool(ord_at4362885an_nat(L),U)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,U),L)) # label(fact_4901_card__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2939 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),zero_zero_real)) -> natfloor(X_1) = zero_zero_nat)) # label(fact_3396_natfloor__neg) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2940 (all Z_1 (is_int(Z_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1)) -> Z_1 = hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat,Z_1))))) # label(fact_3124_nat__0__le) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2941 (all N_57 all A_211 all B_157 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_211),B_157)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_211)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_57)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_211),N_57)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_157),N_57))))))) # label(fact_604_power__strict__mono) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2942 (all A_181 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_181),A_181))) # label(fact_984_dvd__refl) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2943 (all M hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),M) = zero_zero_nat) # label(fact_1638_diff__self__eq__0) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2944 (all K_1 all F (hAPP_nat_nat(F,zero_zero_nat) = zero_zero_nat -> hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),hAPP_n1699378549t_bool(ord_at238088361st_nat(zero_zero_nat),K_1)) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),hAPP_n1699378549t_bool(ord_at238088361st_nat(hAPP_nat_nat(suc,zero_zero_nat)),K_1)))) # label(fact_5080_setsum__shift__lb__Suc0__0) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2945 (all P all Q all R hAPP_b589554111l_bool(P,hAPP_P603027463t_bool(Q,R)) = hAPP_P921862901l_bool(hAPP_f365317245l_bool(hAPP_f1961001761l_bool(cOMBB_1550063917nt_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_016) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2946 (all A_209 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_209),zero_zero_complex) = zero_zero_complex) # label(fact_619_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2947 (all N_1 hAPP_complex_complex(cnj,hAPP_nat_complex(semiri2020571505omplex,N_1)) = hAPP_nat_complex(semiri2020571505omplex,N_1)) # label(fact_4073_complex__cnj__of__nat) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2948 (all L all K_1 (is_int(K_1) & is_int(L) -> (zero_zero_int = K_1 -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),zero_zero_int) = divmod_int(K_1,L)) & (zero_zero_int != K_1 -> (zero_zero_int != L -> divmod_int(K_1,L) = hAPP_P1975530577nt_int(produc713050258nt_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(sgn_sgn_int,L))),hAPP_P1975530577nt_int(hAPP_P1145851913nt_int(hAPP_b40753821nt_int(if_Pro1731782967nt_int,fdisj(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),L)),hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_1)),hAPP_bool_bool(hAPP_b589554111l_bool(fconj,hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,L),zero_zero_int)),hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),zero_zero_int)))),pdivmod(K_1,L)),hAPP_P1975530577nt_int(produc1518849193nt_int(hAPP_f465821005nt_int(hAPP_f1081447804nt_int(cOMBS_1470252563nt_int,hAPP_f617310012nt_int(hAPP_f197974549nt_int(cOMBB_638531587nt_int,cOMBS_1891347093nt_int),hAPP_f1148171384nt_int(hAPP_f334321603nt_int(cOMBB_1484610687nt_int,hAPP_f1582586579nt_int(cOMBC_707974837nt_int,hAPP_f251264923nt_int(hAPP_f279307923nt_int(cOMBB_662120866nt_int,if_Pro1731782967nt_int),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),zero_zero_int)))),hAPP_i1584592887nt_int(hAPP_f465821005nt_int(cOMBC_1964283556nt_int,hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),uminus_uminus_int)),zero_zero_int)))),hAPP_f1574055885nt_int(hAPP_f960630521nt_int(cOMBC_1805044090nt_int,hAPP_f142232355nt_int(hAPP_f1377453843nt_int(cOMBB_320287386nt_int,cOMBB_1846624359nt_int),hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,minus_minus_int),uminus_uminus_int)),one_one_int)))),hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(abs_abs_int,L))))),pdivmod(K_1,L))))) & (L = zero_zero_int -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),K_1) = divmod_int(K_1,L))))) # label(fact_4853_divmod__int__pdivmod) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2949 (all B all A all C (hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C),A) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B),A) <-> C = B)) # label(fact_1937_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2950 (all M_12 hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),hAPP_n522471361de_int(semiri1424489471de_int,M_12)))) # label(fact_2913_zero__le__imp__of__nat) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2951 (all A_90 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_90),one_on1645066479umeral) = A_90) # label(fact_2110_mult__1__right) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2952 (all Z_1 all X_1 all Y_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) -> (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_1),Y_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),Z_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Z_1)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_1),X_1))))) # label(fact_1706_dvd_Ole__less__trans) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2953 (all X_2 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_2),N)),zEven)) <-> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven))))) # label(fact_3151_power__preserves__even) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2954 (all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N)) -> hAPP_int_nat(nat,N) = hAPP_f957591787ol_nat(finite_card_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_eq_int),N)))))) # label(fact_4737_card__bdd__int__set__l__le) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2955 (all X_32 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_32),zero_zero_nat) = one_one_nat) # label(fact_949_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2956 (all B_73 all A_84 all C_36 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_36)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_73),A_84)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_73),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_84),C_36)))))) # label(fact_2156_add__increasing2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2957 (all X_2 hBOOL(hAPP_real_bool(deriv_real(arctan,X_2),hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3878_DERIV__arctan) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2958 (all A_195 all N_47 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_195),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_195),N_47)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_195),N_47)),A_195)) # label(fact_810_power__commutes) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2959 (all Y_2 all X_2 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),Y_2)),zEven))))) # label(fact_2901_even__minus__even) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2960 (all V_16 all W_14 all Z_14 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_16),W_14))),Z_14) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_16)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,W_14)),Z_14))) # label(fact_424_mult__number__of__left) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2961 (all Z_1 hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,Z_1),hAPP_complex_complex(cnj,Z_1)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_complex_real(im,Z_1)))),ii)) # label(fact_4089_complex__diff__cnj) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2962 (all B_93 all A_114 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_114)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_93)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_114),B_93)))))) # label(fact_1881_add__nonneg__pos) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2963 (all A all B ((exists X exists Y (one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A),X)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B),Y)) | one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B),X)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A),Y)))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A),B)))) # label(fact_4035_coprime__bezout) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2964 (all V_10 all U_3 all Y_25 all X_33 all A_185 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_33),A_185)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_25),A_185)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),U_3)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),V_10)) -> (hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,U_3),V_10) = one_one_rat -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,U_3),X_33)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,V_10),Y_25))),A_185)))))))) # label(fact_908_convex__bound__lt) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2965 (all A_22 all C_7 all B_19 (B_19 != zero_z126310315umeral -> hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_7),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_22),B_19)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_22),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_19),C_7))),B_19))) # label(fact_2645_div__mult__self2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2966 (all A all B divmod_int_rel(A,B) = produc93441119t_bool(hAPP_f1048215610t_bool(hAPP_f472159229t_bool(cOMBC_1683390479t_bool,hAPP_f1545556668t_bool(hAPP_f85237525t_bool(cOMBB_389152643ol_int,cOMBS_int_bool_bool),hAPP_f1344929775l_bool(hAPP_f1830214065l_bool(cOMBB_793166024ol_int,hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj)),hAPP_f720654810t_bool(hAPP_f908327869t_bool(cOMBB_1290694363ol_int,hAPP_f1512942609t_bool(cOMBB_int_bool_int,hAPP_i1948725293t_bool(fequal_int,A))),hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,plus_plus_int),hAPP_int_fun_int_int(times_times_int,B)))))),hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_f1805168059t_bool(hAPP_f627970963t_bool(cOMBB_bool_bool_int,fimplies(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))),hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),B))))),hAPP_f1805168059t_bool(hAPP_f627970963t_bool(cOMBB_bool_bool_int,fimplies(hAPP_bool_bool(fNot,hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)))),hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,B))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_eq_int),zero_zero_int))))))) # label(fact_4836_divmod__int__rel__def) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2967 (all C_65 all D_12 all A_131 all B_110 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_131),B_110)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_65),D_12)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_131)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_65)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_131),C_65)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_110),D_12)))))))) # label(fact_1414_mult__strict__mono_H) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2968 (all A_170 all B_145 all K_7 (hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_145),K_7) = A_170 -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,B_145),A_170)))) # label(fact_1099_dvdI) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2969 (all C_72 all B_124 all A_147 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_124),A_147)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_72),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_72),A_147)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_72),B_124)))))) # label(fact_1255_mult__strict__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2970 (all X_2 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),hAPP_real_real(uminus_uminus_real,X_2))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Y_2)),zero_zero_real)))) # label(fact_3491_real__add__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2971 (all B_1_1 all B_2_1 is_bool(hAPP_Q2096512830t_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__Quickcheck____Narrowing__Ocode____int_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2972 (all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sqrt,Y_2))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_2)))) # label(fact_3325_real__sqrt__ge__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2973 (all K_2 hAPP_int_int(bit1,hAPP_int_int(pred,hAPP_int_int(uminus_uminus_int,K_2))) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(bit1,K_2))) # label(fact_4045_minus__Bit1) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2974 (all V_17 all B_160 all C_90 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_17)),B_160)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_17)),C_90)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_17)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_160),C_90))) # label(fact_391_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2975 (all A_3 all B_2 hAPP_int_int(uminus_uminus_int,hAPP_int_int(div_mod_int(A_3),B_2)) = hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,A_3)),hAPP_int_int(uminus_uminus_int,B_2))) # label(fact_3524_zmod__zminus__zminus) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2976 (all V_20 all V_19 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_19)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_20)) -> hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_19)),hAPP_int_rat(number_number_of_rat,V_20)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_19),V_20))))) # label(fact_376_semiring__add__number__of) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2977 (all A_3 all N_1 all B_2 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)),N_1) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N_1)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),N_1))) # label(fact_4978_gcd__exp__int) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2978 (all A_8 all N_14 all B_7 (hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_8),hAPP_nat_nat(suc,N_14)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_7),hAPP_nat_nat(suc,N_14)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_8)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_7)) -> B_7 = A_8)))) # label(fact_3112_power__inject__base) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2979 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),N_1)) -> zero_zero_nat = N_1 | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)))) # label(fact_2287_divides__le) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2980 (all B_127 all A_151 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_151)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_127),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_127),A_151)),zero_zero_int))))) # label(fact_1227_mult__nonneg__nonpos2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2981 (all U (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),U)) -> hAPP_i1948725293t_bool(ord_at641636577an_int(zero_zero_int),U) = image_nat_int(semiri1621563631at_int,hAPP_n1699378549t_bool(ord_lessThan_nat,hAPP_int_nat(nat,U))))) # label(fact_4953_image__atLeastZeroLessThan__int) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2982 (all C_34 all D_6 all A_79 all B_68 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_79),B_68)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_34),D_6)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_79),C_34)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_68),D_6)))))) # label(fact_2206_add__le__less__mono) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2983 (all N_1 all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> N_1 = one_one_nat | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),N_1)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,P_3),N_1)))) # label(fact_4187_prime__coprime) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2984 (all X_2 all Y_2 (is_int(Y_2) & is_int(X_2) -> (X_2 = zero_zero_int & Y_2 = zero_zero_int <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_2),X_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_2),Y_2))),zero_zero_int))))) # label(fact_731_sum__squares__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2985 (all P all Q all R hAPP_rat_bool(P,hAPP_int_rat(Q,R)) = hAPP_int_bool(hAPP_f1241350704t_bool(hAPP_f1131424617t_bool(cOMBB_rat_bool_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Rat__Orat_000tc__HOL__Obool_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2986 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(ln,X_1)),X_1)))) # label(fact_3428_ln__less__self) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2987 (all A_223 (is_int(A_223) -> A_223 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,pls)),A_223))) # label(fact_413_add__numeral__0) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2988 (all A_72 all B_64 all C_32 hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_72),B_64)),C_32) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_72),hAPP_nat_nat(div_mod_nat(B_64),C_32))),C_32)) # label(fact_2315_mod__mult__right__eq) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2989 (all A_133 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_133),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_133),one_one_real)))) # label(fact_1370_less__add__one) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2990 (all Y_2 all X_2 all A (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(log(A),X_2)),hAPP_real_real(log(A),Y_2)))))))) # label(fact_3866_log__less__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2991 (all X_1 all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(standardRes(P_3),X_1)),P_3)))) # label(fact_2445_StandardRes__ubound) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2992 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),zero_zero_int))))) # label(fact_2629_div__nonneg__neg__le0) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2993 (all A all C all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B),C))))) # label(fact_2033_add__le__cancel__right) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2994 (all K_2 hAPP_int_rat(number_number_of_rat,K_2) = frct(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,K_2)),one_one_int))) # label(fact_4640_Frct__code__post_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2995 (all X_47 all P_8 all Q_12 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_47),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_8),Q_12)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_47),P_8)),Q_12)) # label(fact_71_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2996 (all A (zero_zero_nat = A <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A)))) # label(fact_2275_gcd__lcm__complete__lattice__nat_Otop__unique) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2997 (all A_229 all B_162 all C_91 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_229),C_91)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_162),C_91)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_229),B_162)),C_91)) # label(fact_312_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2998 (all B_2 all D_3 all A_3 (one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),A_3) -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),B_2) = one_one_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2)) = one_one_nat))) # label(fact_4354_coprime__mult__nat) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 2999 (all X_1 all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_1)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_3360_real__less__half__sum) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3000 (all B_97 all A_118 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_118),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_97),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_118),B_97)),zero_zero_int))))) # label(fact_1857_add__nonpos__neg) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3001 (all X_8 all Q_6 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_8),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_8),Q_6)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_8),hAPP_nat_nat(suc,Q_6))) # label(fact_3017_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3002 (all F all N all R_2 hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,minus_minus_real),F)),R_2)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),R_2))) # label(fact_4896_sumr__diff__mult__const) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3003 (all X_2 hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),cos)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_2),hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(cos,X_2))),hAPP_real_real(sin,X_2)))))) # label(fact_4797_DERIV__cos__realpow2b) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3004 (all A_18 all B_15 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(div_mod_nat(A_18),B_15)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_18),B_15)),B_15)) = A_18) # label(fact_2668_semiring__div__class_Omod__div__equality_H) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3005 (all K_1 all I_1 all J hAPP_n1699378549t_bool(ord_at238088361st_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),K_1)) = image_nat_nat(hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),K_1),hAPP_n1699378549t_bool(ord_at238088361st_nat(I_1),J))) # label(fact_5072_image__add__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3006 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),zero_zero_int))))) # label(fact_2626_div__nonpos__pos__le0) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3007 (all B_129 all A_153 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_153),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_129),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_153),B_129)))))) # label(fact_1218_mult__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3008 (all B_1_1 all B_2_1 is_int(hAPP_f1594865479ol_int(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__Int__Oint) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3009 (all Y_2 the_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,ord_less_eq_real),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),sin)),Y_2)))) = hAPP_real_real(arcsin,Y_2)) # label(fact_4882_arcsin__def) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3010 (all N_54 all A_204 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_204)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_204),N_54))))) # label(fact_662_zero__less__power) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3011 (all A_172 all B_147 all C_85 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_172),B_147)),C_85)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,B_147),C_85)))) # label(fact_1086_dvd__mult__right) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3012 (all P_1 all L ((exists X hBOOL(hAPP_C24501650l_bool(P_1,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,L),X)))) <-> (exists X (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,L),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X),zero_z126310315umeral))) & hBOOL(hAPP_C24501650l_bool(P_1,X)))))) # label(fact_1806_unity__coeff__ex) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3013 (all A_105 A_105 = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,zero_z126310315umeral),A_105)) # label(fact_1980_add__0) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3014 (all Ma all K_1 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),K_1))) <-> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N))))) # label(fact_1824_mult__le__cancel2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3015 (all K_2 hAPP_int_int(pred,hAPP_int_int(bit1,K_2)) = hAPP_int_int(bit0,K_2)) # label(fact_4039_pred__Bit1) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3016 (all A_230 all M_29 all B_163 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_230),M_29)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_163),M_29)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_230),B_163)),M_29)) # label(fact_307_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3017 (all C_74 all A_155 all B_131 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_155),B_131)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_74)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_74),A_155)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_74),B_131)))))) # label(fact_1206_mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3018 (all P all Q all R hAPP_real_real(hAPP_r1250527377l_real(P,R),Q) = hAPP_real_real(hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__RealDef_) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3019 (all C_69 all A_144 all B_121 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_144),B_121)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_69)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_69),A_144)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_69),B_121)))))) # label(fact_1271_mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3020 (all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,pls)),Y_2)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,one_one_rat),hAPP_int_rat(number_number_of_rat,Y_2))))) # label(fact_295_le__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3021 (all A_34 A_34 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_34),one_one_nat)) # label(fact_2546_div__by__1) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3022 (all B_1_1 all B_2_1 all B_3_1 is_bool(tendsto_complex_real(B_1_1,B_2_1,B_3_1))) # label(gsy_c_Limits_Otendsto_000tc__Complex__Ocomplex_000tc__RealDef__Oreal) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3023 (all M_10 all N_24 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_10),N_24)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_10)),hAPP_nat_int(semiri1621563631at_int,N_24))))) # label(fact_2942_less__imp__of__nat__less) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3024 (all P all Q (hBOOL(Q) | hBOOL(P) | -hBOOL(fdisj(P,Q)))) # label(help_fdisj_3_1_U) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3025 (all A_88 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,one_one_real),A_88) = A_88) # label(fact_2123_mult__1__left) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3026 (all A_3 all B_2 hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,B_2),A_3) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)) # label(fact_4979_gcd__commute__int) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3027 (all X_30 all N_34 (one_one_nat = X_30 | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_34)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_30),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_30),N_34))))) # label(fact_998_dvd__power) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3028 (all A_238 all B_167 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_167),A_238) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_238),B_167)) # label(fact_186_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3029 (all F all A_1 hBOOL(hAPP_f54304608l_bool(nat_tr160667106at_int(F),A_1))) # label(fact_2786_transfer__morphismI) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3030 (all N_1 zero_zero_real != hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1))) # label(fact_3761_real__of__nat__fact__not__zero) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3031 (all P_1 collec1347809874nt_int(P_1) = P_1) # label(fact_209_Collect__def) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3032 (all R_3 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,R_3),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,archim1246769320r_real(R_3))),one_one_real)))) # label(fact_4450_real__of__int__floor__add__one__ge) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3033 (all X_43 all Y_35 all Q_10 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_43),Q_10)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_35),Q_10)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_43),Y_35)),Q_10)) # label(fact_347_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3034 (all Y_6 all P_6 all N_10 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_6),N_10)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),P_6))),Y_6) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N_10)),P_6)))) # label(fact_3179_lemma__realpow__diff) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3035 (all B_115 all A_138 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_138)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_115),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_115),A_138)),zero_zero_int))))) # label(fact_1302_mult__pos__neg2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3036 (all X_2 hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),cos)),cos),X_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,min))),hAPP_real_real(cos,X_2))),hAPP_real_real(sin,X_2))))) # label(fact_4781_DERIV__cos__cos__mult2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3037 (all M all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),J_2))))) # label(fact_1778_trans__le__add2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3038 (all A_184 hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_184),zero_z891286103de_int))) # label(fact_914_dvd__0__right) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3039 (exists K_4 all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(im,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X)),K_4)))) # label(fact_4108_Im_Obounded) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3040 (all G all X_2 (hBOOL(hAPP_real_bool(sums_real(G),X_2)) -> hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_r195310020l_real(hAPP_f2126667875l_real(cOMBC_746811033l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),zero_zero_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,G),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,div_div_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,minus_minus_nat),one_one_nat))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),X_2)))) # label(fact_4867_sums__if_H) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3041 (all B all A all D (is_int(D) -> (hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A),B) = D <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),A)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),B)) & (all E (is_int(E) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,E),A)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,E),B)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,E),D))))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),D))))) # label(fact_5011_gcd__unique__int) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3042 (all N_1 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),N_1))))))) # label(fact_3816_Ln_Oaux1) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3043 (all A_236 all B_165 all C_96 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_236),C_96)),B_165) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_236),B_165)),C_96)) # label(fact_218_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3044 (all K_2 hAPP_int_int(succ,hAPP_int_int(bit1,K_2)) = hAPP_int_int(bit0,hAPP_int_int(succ,K_2))) # label(fact_3297_succ__Bit1) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3045 (all C_60 all A_126 all B_105 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_60),A_126)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_60),B_105))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_60)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_126),B_105))))) # label(fact_1444_mult__left__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3046 (all C_60 all A_126 all B_105 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_60),A_126)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_60),B_105))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_60)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_126),B_105))))) # label(fact_1443_mult__left__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3047 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(d22set,A)))))) # label(fact_2273_d22set__mem) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3048 (all X_1 all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(multInv(P_3),hAPP_int_int(multInv(P_3),X_1))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),hAPP_int_int(multInv(P_3),X_1))),hAPP_int_int(multInv(P_3),hAPP_int_int(multInv(P_3),X_1)))),P_3)))))) # label(fact_2225_Int2_Oaux____1) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3049 (all Ta all A_1 all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1)) -> (all X (is_int(X) -> ((all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A_1)) -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) != X)))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ta),X)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ta),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),D_1))))))))) # label(fact_5094_aset_I7_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3050 (all F all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)) -> ((all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),B)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X)) -> (exists Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y)) & hBOOL(hAPP_real_bool(deriv_real(F,X),Y)))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,A)),hAPP_real_real(F,B)))))) # label(fact_4002_DERIV__pos__imp__increasing) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3051 (all A_221 A_221 = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(bit1,pls))),A_221)) # label(fact_454_mult__numeral__1) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3052 (all P_3 quotient_of(hAPP_rat_rat(sgn_sgn_rat,P_3)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(sgn_sgn_int,hAPP_P1175774780nt_int(product_fst_int_int,quotient_of(P_3)))),one_one_int)) # label(fact_4652_rat__sgn__code) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3053 (all P all Q all R hAPP_bool_bool(hAPP_i68813070l_bool(P,R),hAPP_int_bool(Q,R)) = hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__HOL__Obool_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3054 (all M_6 all N_20 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(semiri984289939at_nat,M_6)),N_20) = hAPP_nat_nat(semiri984289939at_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_6),N_20))) # label(fact_2983_of__nat__power) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3055 (all Q_2 all B_2 all R_3 all C_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),R_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_3),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),C_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(div_mod_int(Q_2),C_1))),R_3))))))) # label(fact_2465_zmult2__lemma__aux1) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3056 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_2)) -> hAPP_int_int(div_mod_int(A_3),B_2) = hAPP_P1175774780nt_int(product_snd_int_int,posDivAlg(A_3,B_2))))) # label(fact_4234_mod__pos__pos) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3057 (all X_1 all M hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,X_1)),M) = hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,hAPP_int_int(div_mod_int(X_1),M))),M)) # label(fact_3525_zminus__zmod) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3058 (all A_105 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A_105) = A_105) # label(fact_1986_add__0) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3059 (all A_186 all K_8 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_186),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),K_8))),zero_zero_real)) -> A_186 = zero_zero_real)) # label(fact_900_even__power__le__0__imp__0) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3060 (all C_83 all D_16 all A_169 all B_144 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_169),B_144)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C_83),D_16)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_169),C_83)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_144),D_16)))))) # label(fact_1112_mult__dvd__mono) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3061 (all X_2 all Y_2 (is_int(Y_2) & is_int(X_2) -> (hAPP_int_int(number_number_of_int,Y_2) = hAPP_int_int(number_number_of_int,X_2) <-> Y_2 = X_2))) # label(fact_247_eq__number__of) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3062 (all X_29 all Y_23 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_29),Y_23)) -> (Y_23 != X_29 -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_29),Y_23))))) # label(fact_1017_order__le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3063 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),N_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N_1),M)) -> M = N_1))) # label(fact_1702_dvd__antisym) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3064 (all A_173 (is_int(A_173) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,zero_zero_int),A_173)) -> A_173 = zero_zero_int))) # label(fact_1083_dvd__0__left) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3065 (all A_64 all B_56 hAPP_nat_nat(div_mod_nat(A_64),B_56) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_64),B_56)),B_56)) # label(fact_2343_mod__add__self2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3066 (all P all Q all R hAPP_r1134773055l_bool(hAPP_f1884005689l_bool(hAPP_f172451459l_bool(cOMBB_421479067l_real,P),Q),R) = hAPP_f330985925l_bool(P,hAPP_r2019696015l_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__prod_Itc__RealDef__Orea) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3067 (all V_2 ((hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> hAPP_nat_real(real_nat,hAPP_int_nat(number_number_of_nat,V_2)) = zero_zero_real) & (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> hAPP_int_real(number267125858f_real,V_2) = hAPP_nat_real(real_nat,hAPP_int_nat(number_number_of_nat,V_2))))) # label(fact_3699_real__of__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3068 (all X_1 all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natfloor(X_1)),natfloor(Y_1))))) # label(fact_3385_natfloor__mono) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3069 (all C_74 all A_155 all B_131 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_155),B_131)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_74)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_74),A_155)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_74),B_131)))))) # label(fact_1209_mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3070 (all P all Q all R hAPP_f1509287679l_real(P,hAPP_r474017924t_real(Q,R)) = hAPP_r1409264866l_real(hAPP_f1523861863l_real(hAPP_f1067681227l_real(cOMBB_1934313333l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_000tc__fun__022) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3071 (all A_181 hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_181),A_181))) # label(fact_979_dvd__refl) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3072 (all N all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),Ma)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(fact_fact_nat,Ma)),hAPP_nat_nat(fact_fact_nat,N)) = big_co1705425894at_nat(cOMBI_nat,hAPP_n1699378549t_bool(ord_at238088361st_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),Ma)))) # label(fact_5082_fact__div__fact) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3073 (all X_6 all Y_5 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_6),Y_5)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_6),Y_5)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_6),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_5),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))))) # label(fact_3192_realpow__two__diff) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3074 (all X_2 (is_int(X_2) -> ((exists A_2 exists B_4 (hAPP_P1175774780nt_int(twoSqu1907779896sum2sq,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_2),B_4)) = X_2 & is_int(B_4) & is_int(A_2))) <-> hBOOL(hAPP_int_bool(twoSqu1770640450sum2sq,X_2))))) # label(fact_1635_is__sum2sq__def) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3075 (all I_1 all J all F ((all I_2 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,I_2)),hAPP_nat_nat(F,J_1))))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,I_1)),hAPP_nat_nat(F,J)))))) # label(fact_2816_less__mono__imp__le__mono) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3076 (all C_34 all D_6 all A_79 all B_68 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_79),B_68)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_34),D_6)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_79),C_34)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_68),D_6)))))) # label(fact_2205_add__le__less__mono) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3077 (all N all X_2 exists T (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_2))) & hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_r195310020l_real(hAPP_f2126667875l_real(cOMBC_746811033l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),zero_zero_real)),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,div_div_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,minus_minus_nat),hAPP_nat_nat(suc,zero_zero_nat)))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))))),hAPP_r474017924t_real(power_power_real,X_2))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,T),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(real_nat,N))),pi)))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),N))) = hAPP_real_real(sin,X_2))) # label(fact_4913_Maclaurin__sin__expansion2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3078 (all P all Q all R hAPP_r2019696015l_real(hAPP_f1591894897l_real(hAPP_f644105375l_real(cOMBC_467724138l_real,P),Q),R) = hAPP_f617735571l_real(hAPP_r712953929l_real(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__R) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3079 (all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(nat,N)),one_one_nat) = hAPP_f957591787ol_nat(finite_card_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_eq_int),N)))))) # label(fact_4773_card__bdd__int__set__le) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3080 (all A_110 all B_89 all C_49 (hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_110),B_89) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_110),C_49) -> C_49 = B_89)) # label(fact_1921_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3081 (all V_15 all W_13 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_15)),hAPP_int_complex(number528085621omplex,W_13)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_15),W_13))) # label(fact_427_arith__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3082 (all A_194 all B_153 all N_46 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_194),N_46)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_153),N_46)) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_194),B_153)),N_46)) # label(fact_812_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3083 (all B_12 all A_15 all C_5 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_15),C_5) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_12),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_15),B_12))),hAPP_C498520661umeral(div_mo1740067990umeral(A_15),B_12))),C_5)) # label(fact_2681_div__mod__equality2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3084 (all X_32 one_on1645066479umeral = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_32),zero_zero_nat)) # label(fact_951_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3085 (all P all Q all R hAPP_P1975530577nt_int(P,hAPP_i1524277240nt_int(Q,R)) = hAPP_i1524277240nt_int(hAPP_f1735705551nt_int(hAPP_f151095827nt_int(cOMBB_1389251822nt_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__prod_Itc) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3086 (all C_68 all A_143 all B_120 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_143),B_120)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_68)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_143),C_68)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_120),C_68)))))) # label(fact_1274_mult__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3087 (all A (is_int(A) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) <-> A != zero_zero_int))) # label(fact_798_zero__less__power2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3088 (all X_1 all M hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,M)),pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,M))),pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_3694_sin__expansion__lemma) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3089 (all X_2 all A_1 (hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A_1)) -> hBOOL(hAPP_int_bool(nat_is_nat,X_2))))) # label(fact_5138_Nat__Transfer_Otransfer__int__nat__set__function__closures_I6_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3090 (all A_58 all C_21 all B_50 hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_58),B_50)),C_21) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_58),C_21)),B_50)),C_21)) # label(fact_2369_zmod__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3091 (all X_21 all N_31 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_21),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_31),one_one_nat))),X_21) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_21),N_31))) # label(fact_1738_realpow__minus__mult) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3092 (all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> hAPP_nat_nat(fact_fact_nat,N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),one_one_nat))))) # label(fact_3815_fact__reduce__nat) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3093 (all X_1 all M hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),hAPP_int_int(standardRes(M),X_1)),M))) # label(fact_2406_StandardRes__prop1) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3094 (all A_3 -(-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),zero_zero_nat)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A_3)))) # label(fact_2278_gcd__lcm__complete__lattice__nat_Onot__top__less) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3095 (all Y_1 all N_1 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_1),N_1))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),Y_1)))))) # label(fact_964_zpower__zdvd__prop2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3096 (all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> (one_one_real != A_3 -> hAPP_real_real(log(A_3),A_3) = one_one_real))) # label(fact_3867_log__eq__one) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3097 (all B all A (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B),A) = B <-> A = zero_zero_real)) # label(fact_638_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3098 (all X_1 all Y_1 all Z_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Y_1)),Z_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y_1),Z_1))) # label(fact_4311_gcd__semilattice__nat_Oinf__assoc) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3099 (all X_2 all Y_2 (-hBOOL(hAPP_int_bool(even_odd_even_int,X_2)) & -hBOOL(hAPP_int_bool(even_odd_even_int,Y_2)) | hBOOL(hAPP_int_bool(even_odd_even_int,Y_2)) & hBOOL(hAPP_int_bool(even_odd_even_int,X_2)) <-> hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),Y_2))))) # label(fact_2894_even__difference) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3100 (all V all V_1 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls)) -> hAPP_int_nat(number_number_of_nat,V) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_int_nat(number_number_of_nat,V_1))) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_int_nat(number_number_of_nat,V_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V),hAPP_int_int(uminus_uminus_int,V_1))))),zero_zero_nat),hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V),hAPP_int_int(uminus_uminus_int,V_1)))))))) # label(fact_4795_diff__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3101 (all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arctan,Y_1))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arctan,Y_1)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_3501_arctan__bounded) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3102 (all C_73 all A_154 all B_130 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_154),B_130)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_73)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_154),C_73)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_130),C_73)))))) # label(fact_1213_mult__right__mono) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3103 (all D all Sa exists Z all X (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,X),Z)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,D),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X),Sa))) <-> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,D),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X),Sa)))))) # label(fact_2251_minf_I9_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3104 (all Ta all D all D_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),D_1)) -> (all X all K (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,K),D_1))),Ta))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Ta))))))) # label(fact_1813_inf__period_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3105 (all P all Q all R hAPP_i1796122964t_bool(hAPP_f1902329768t_bool(hAPP_f1182713365t_bool(cOMBB_2087181719ol_int,P),Q),R) = hAPP_f561022312t_bool(P,hAPP_i614188481l_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__fun_Itc__HOL__Obool_Mtc__HOL) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3106 (all A (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),A))))) # label(fact_2182_zero__le__double__add__iff__zero__le__single__add) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3107 (all Y_1 all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> (one_one_real != A_3 -> hAPP_real_real(log(A_3),hAPP_real_real(hAPP_r1250527377l_real(powr,A_3),Y_1)) = Y_1))) # label(fact_4460_log__powr__cancel) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3108 (all N_1 all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(nat,X_1)),N_1) = hAPP_int_nat(nat,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),N_1)))) # label(fact_2760_Nat__Transfer_Otransfer__nat__int__functions_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3109 (all K_2 all I all J_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),J_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I),K_2)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,J_2),K_2))))) # label(fact_120_zadd__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3110 (all Y_15 zero_zero_complex = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,zero_zero_complex),Y_15)) # label(fact_2473_mult__left_Ozero) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3111 (all P all Q all R hAPP_real_bool(hAPP_f101635158l_bool(hAPP_f1925507189l_bool(cOMBB_1121934354l_real,P),Q),R) = hAPP_f448129468l_bool(P,hAPP_r481142414t_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__HOL__Obo_006) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3112 (all K_1 all Ma all N (zero_zero_nat = K_1 | Ma = N <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N))) # label(fact_1842_nat__mult__eq__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3113 (all Ma all K_1 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),K_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),N)) | K_1 = zero_zero_nat)) # label(fact_1612_nat__mult__dvd__cancel__disj_H) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3114 (all Nat_2 hAPP_nat_nat(suc,Nat_2) != zero_zero_nat) # label(fact_2854_nat_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3115 (all A_171 all B_146 all C_84 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_171),B_146)),C_84)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_171),C_84)))) # label(fact_1097_dvd__mult__left) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3116 (all P all Q all R hAPP_real_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,P),Q),R) = hAPP_nat_real(hAPP_r474017924t_real(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__Nat__Onat_000tc__RealDef__Orea) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3117 (all X_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> X_1 = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),N_1)))) # label(fact_3935_real__root__pow__pos) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3118 (all X_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,one_one_nat),X_1) = one_one_nat) # label(fact_4334_gcd__lcm__complete__lattice__nat_Oinf__bot__left) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3119 (all R1 all A_3 all R2 all B_2 rcis(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R1),R2),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_3),B_2)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,rcis(R1,A_3)),rcis(R2,B_2))) # label(fact_4263_rcis__mult) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3120 (all A_221 A_221 = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit1,pls))),A_221)) # label(fact_455_mult__numeral__1) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3121 (all Y_1 all X_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) -> Y_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Y_1))) # label(fact_4321_gcd__proj2__if__dvd__nat) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3122 (all W hAPP_int_int(uminus_uminus_int,hAPP_int_int(number_number_of_int,W)) = hAPP_int_int(number_number_of_int,hAPP_int_int(uminus_uminus_int,W))) # label(fact_3515_minus__numeral__code_I5_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3123 (all X_1 all Y_1 hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_complex_real(norm_norm_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,X_1),Y_1))) # label(fact_3953_complex__norm) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3124 (all A_210 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,zero_zero_real),A_210) = zero_zero_real) # label(fact_610_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3125 (all C_42 all A_99 all B_82 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_99),B_82)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_99),C_42)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_82),C_42))))) # label(fact_2020_add__right__mono) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3126 (all Z_1 all W hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_1),W) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),hAPP_int_int(uminus_uminus_int,W))) # label(fact_3536_diff__int__def__symmetric) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3127 (all U all Ma all N all I_1 all J (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),I_1)),U)),N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U)),N)))))) # label(fact_1849_nat__le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3128 (all A (zero_zero_rat = A <-> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_rat)) # label(fact_779_zero__eq__power2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3129 (all B_128 all A_152 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_152),zero_z891286103de_int)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_128)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_152),B_128)),zero_z891286103de_int))))) # label(fact_1225_mult__nonpos__nonneg) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3130 (all V_2 hAPP_int_real(number267125858f_real,V_2) = hAPP_complex_real(re,hAPP_int_complex(number528085621omplex,V_2))) # label(fact_4124_complex__Re__number__of) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3131 (all X_2 all Y_2 (zero_zero_real = Y_2 & zero_zero_real = X_2 <-> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_2),X_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_2),Y_2)) = zero_zero_real)) # label(fact_691_sum__squares__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 5.65/5.67 3132 (all B_118 all A_141 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_141),zero_zero_rat)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_118)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_141),B_118)),zero_zero_rat))))) # label(fact_1286_mult__neg__pos) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3133 (all Ta all B_1 all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1)) -> (all X (is_int(X) -> ((all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B_1)) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) != X)))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),Ta)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),D_1)),Ta)))))))) # label(fact_5095_bset_I5_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3134 (all X_1 all Y_1 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(re,X_1)),hAPP_complex_real(re,Y_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(im,X_1)),hAPP_complex_real(im,Y_1))) = hAPP_complex_real(re,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_1),Y_1))) # label(fact_4130_complex__Re__mult) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3135 (all X_24 all Y_18 all A_120 all B_100 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_24),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,Y_18),B_100))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_24),A_120)),B_100)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_24),Y_18)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_120),B_100))) # label(fact_1542_mult__diff__mult) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3136 (all B_119 all A_142 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_142),zero_zero_rat)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_119),zero_zero_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_142),B_119)))))) # label(fact_1280_mult__neg__neg) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3137 (all X_1 hAPP_int_real(number267125858f_real,X_1) = hAPP_int_real(ring_1_of_int_real,X_1)) # label(fact_4580_number__of__real__def) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3138 (all A_110 all B_89 all C_49 (hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_110),B_89) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_110),C_49) -> B_89 = C_49)) # label(fact_1917_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3139 (all K_1 all P_1 all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D)) -> ((all X (is_int(X) -> (hBOOL(hAPP_int_bool(P_1,X)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),D)))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_1)) -> (all X (hBOOL(hAPP_int_bool(P_1,X)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),D)))))))))) # label(fact_1828_incr__mult__lemma) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3140 (all N_1 all M hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),M)),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,N_1),M)) # label(fact_5187_nat__minus__add__max) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3141 (all A_3 all B_2 all C_1 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(div_mod_int(B_2),C_1))),C_1) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)),C_1)) # label(fact_2396_zmod__zmult1__eq) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3142 (all X_2 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_2),Y_2)),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Y_2)))) # label(fact_1680_real__le__eq__diff) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3143 (all X_1 hAPP_real_real(tan,X_1) = hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),pi))) # label(fact_3599_tan__periodic__pi) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3144 (all A_119 all B_98 all V_6 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_119),B_98)),hAPP_int_real(number267125858f_real,V_6)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_119),hAPP_int_real(number267125858f_real,V_6))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_98),hAPP_int_real(number267125858f_real,V_6)))) # label(fact_1560_left__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3145 (all B_2 zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,zero_zero_int),B_2)) # label(fact_2536_zdiv__zero) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3146 (all B_2 all A_3 (is_int(A_3) & is_int(B_2) -> (zero_zero_int != A_3 | B_2 != zero_zero_int -> one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)))))) # label(fact_4966_div__gcd__coprime__int) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3147 (all X_43 all Y_35 all Q_10 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_43),Y_35)),Q_10) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_43),Q_10)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_35),Q_10))) # label(fact_343_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3148 (all A_112 all B_91 all C_51 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_112),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_91),C_51)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_112),B_91)),C_51)) # label(fact_1909_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3149 (all C_44 all D_8 all A_101 all B_84 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_101),B_84)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_44),D_8)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_101),C_44)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_84),D_8)))))) # label(fact_2008_add__mono) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3150 (all B_69 all A_80 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_80)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_69)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_80),B_69)))))) # label(fact_2192_add__pos__pos) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3151 (all M hAPP_int_int(legacy_zgcd(zero_zero_int),M) = hAPP_int_int(abs_abs_int,M)) # label(fact_4503_zgcd__0__left) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3152 (all M all N_1 all S_3 all S_1 all T_2 all T_1 all R_5 all R_3 all K_2 (hAPP_int_int(legacy_zgcd(R_5),R_3) = K_2 -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),R_3)) -> (exists Sn exists Tn (is_int(Sn) & xzgcda(M,N_1,R_5,R_3,S_3,S_1,T_2,T_1) = hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,K_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Sn),Tn)) & is_int(Tn)))))) # label(fact_4534_xzgcd__correct__aux1) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3153 (all B_2 all D_3 all A_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),A_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_3),B_2))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),B_2))))) # label(fact_1594_divides__add__revr) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3154 (all N big_co1705425894at_nat(cOMBI_nat,hAPP_n1699378549t_bool(ord_at238088361st_nat(one_one_nat),N)) = hAPP_nat_nat(fact_fact_nat,N)) # label(fact_5073_fact__altdef__nat) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3155 (all A_197 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_197),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_780_zero__le__power2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3156 (all A_166 all B_141 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_166),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_141),A_166)))) # label(fact_1132_dvd__triv__right) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3157 (all M all N_1 all K_2 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_1)))) # label(fact_4299_gcd__dvd__prod__nat) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3158 (all X_37 all Y_29 (hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_37),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_29),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),X_37)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),Y_29)) -> Y_29 = X_37)))) # label(fact_793_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3159 (all N_1 all A_3 all B_2 (hAPP_int_int(legacy_zgcd(A_3),B_2) = one_one_int -> one_one_int = hAPP_int_int(legacy_zgcd(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N_1)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),N_1)))) # label(fact_4506_zgcd__1__power__distrib) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3160 (all A all E_2 all C all B all D (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,B),A)),E_2)),D))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),E_2)),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),E_2)),D))))) # label(fact_1577_less__add__iff2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3161 (all A_230 all M_29 all B_163 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_230),B_163)),M_29) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_230),M_29)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_163),M_29))) # label(fact_306_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3162 (all X_1 hAPP_complex_complex(cnj,hAPP_complex_complex(uminus473333897omplex,X_1)) = hAPP_complex_complex(uminus473333897omplex,hAPP_complex_complex(cnj,X_1))) # label(fact_4069_cnj_Ominus) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3163 (all M_25 all N_65 all A_216 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_216)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_216),M_25)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_216),N_65))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_25),N_65))))) # label(fact_528_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3164 (all N all X_2 exists T (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_2))) & hAPP_real_real(cos,X_2) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f451358433l_real(hAPP_f335634176l_real(cOMBS_337378985l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,div_div_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),zero_zero_real))),hAPP_r474017924t_real(power_power_real,X_2))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,T),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(real_nat,N))),pi)))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),N))))) # label(fact_4914_Maclaurin__cos__expansion) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3165 (all X_3 ((all R_1 exists N_3 all N_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_3),N_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_1),hAPP_nat_real(X_3,N_2))))) -> hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),X_3),zero_zero_real,sequentially)))) # label(fact_5105_LIMSEQ__inverse__zero) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3166 (all V_17 all B_160 all C_90 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_i1732201573de_int(number1226105091de_int,V_17)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_160),C_90)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_i1732201573de_int(number1226105091de_int,V_17)),B_160)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_i1732201573de_int(number1226105091de_int,V_17)),C_90))) # label(fact_390_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3167 (all X_1 hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,pi),X_1)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_1))) # label(fact_3669_sin__periodic__pi2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3168 (all X_1 all Y_1 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_real_real(uminus_uminus_real,Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),Y_1)) # label(fact_3486_minus__real__def) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3169 (all A_3 all Q_2 all R_3 (is_int(A_3) & is_int(R_3) -> (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,A_3),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3))) -> (A_3 != zero_zero_int -> zero_zero_int = R_3)))) # label(fact_2791_self__remainder) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3170 (all B_1 all A_1 (hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1)) -> (hBOOL(hAPP_f448129468l_bool(nat_nat_set,B_1)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,image_int_nat(nat,A_1)),image_int_nat(nat,B_1))) <-> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,A_1),B_1)))))) # label(fact_4610_transfer__int__nat__set__relations_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3171 (all A_227 A_227 = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,one_one_complex),A_227)) # label(fact_338_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3172 (all X_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_complex_real(re,X_1)),hAPP_complex_real(norm_norm_complex,X_1)))) # label(fact_4117_complex__Re__le__cmod) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3173 (all A_104 (is_int(A_104) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),A_104) = A_104)) # label(fact_1988_add__0__left) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3174 (all X_44 all Y_36 all Z_15 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_44),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Y_36),Z_15)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_44),Y_36)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_44),Z_15))) # label(fact_323_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3175 (all A_3 all B_2 (zero_zero_nat != B_2 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),B_2)))) # label(fact_4344_gcd__le2__nat) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3176 (all X_2 all Y_2 all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),B)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B),X_2)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B),Y_2)))))) # label(fact_880_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3177 (all X_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),pls)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,X_2)),zero_zero_real)))) # label(fact_766_le__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3178 (all P_1 all K_1 all I_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),I_1)) -> (hBOOL(hAPP_int_bool(P_1,K_1)) -> ((all I_2 (is_int(I_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),I_2)) -> (hBOOL(hAPP_int_bool(P_1,I_2)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I_2),one_one_int))))))) -> hBOOL(hAPP_int_bool(P_1,I_1)))))) # label(fact_541_int__ge__induct) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3179 (all X_2 all N all Diff all F (hAPP_n546711566l_real(Diff,zero_zero_nat) = F -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (zero_zero_real != X_2 -> ((all M_1 all X hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),X),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),X)))) -> (exists T (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,X_2))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),N))) = hAPP_real_real(F,X_2) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_2))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(abs_abs_real,T)))))))))) # label(fact_4930_Maclaurin__all__lt) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3180 (all M hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(M),zero_zero_int),M))) # label(fact_1158_zcong__id) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3181 (all N hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,hAPP_nat_real(real_nat,N)),ring_1_Ints_real))) # label(fact_4758_Ints__real__of__nat) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3182 (all Ma all No all F ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,N_2)),hAPP_nat_real(F,hAPP_nat_nat(suc,N_2))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,Ma)),hAPP_nat_real(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),No)))))) # label(fact_3373_lemma__f__mono__add) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3183 (all P all Q all R hAPP_real_real(P,hAPP_real_real(Q,R)) = hAPP_real_real(hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__RealDef_) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3184 (all L all U hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(ord_gr1297742076an_int(L),U)))) # label(fact_4628_finite__greaterThanLessThan__int) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3185 (all P all Q all R hAPP_r847664445t_bool(hAPP_f1920544743t_bool(hAPP_f866672739t_bool(cOMBB_1247074793l_real,P),Q),R) = hAPP_f1173902867t_bool(P,hAPP_r1134773055l_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__HOL__Obool_J_000tc__fun) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3186 (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X_2)),one_one_real)) -> hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,one_one_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_2)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat)))))))) # label(fact_4826_summable__arctan__series) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3187 (all C_64 all D_11 all A_130 all B_109 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_130),B_109)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_64),D_11)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_130)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_64)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_130),C_64)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_109),D_11)))))))) # label(fact_1424_mult__less__le__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3188 (all C_57 all A_123 all B_102 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_57),A_123)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_57),B_102))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_57)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_123),B_102))))) # label(fact_1465_mult__left__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3189 (all C_89 all D_17 all A_200 all B_154 all R_8 (zero_zero_real != R_8 -> (B_154 = A_200 & C_89 != D_17 -> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_154),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_8),D_17)) != hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_200),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_8),C_89))))) # label(fact_695_add__scale__eq__noteq) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3190 (all A all B all C (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),A)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),B))))) # label(fact_1401_mult__le__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3191 (all N all F (hBOOL(summable_real(F)) -> ((all M_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(F,M_1))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_f352196356l_real(suminf_real,F)))))) # label(fact_4929_series__pos__le) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3192 (all C_34 all D_6 all A_79 all B_68 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_79),B_68)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,C_34),D_6)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_79),C_34)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_68),D_6)))))) # label(fact_2202_add__le__less__mono) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3193 (all Y_1 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X_1)),one_one_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,Y_1)),one_one_real)) -> hAPP_real_real(arctan,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_1)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),Y_1)))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(arctan,X_1)),hAPP_real_real(arctan,Y_1))))) # label(fact_3407_arctan__add) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3194 (all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),B)),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),zero_zero_real)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B),zero_zero_real)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A)))) # label(fact_1246_mult__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3195 (all P_1 all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D)) -> ((all X all K (is_int(K) & is_int(X) -> (hBOOL(hAPP_int_bool(P_1,X)) <-> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D))))))) -> ((exists X (is_int(X) & hBOOL(hAPP_int_bool(P_1,X)) & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D))))) <-> (exists X1 (hBOOL(hAPP_int_bool(P_1,X1)) & is_int(X1))))))) # label(fact_5100_periodic__finite__ex) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3196 (all C all A all B (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C),A)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C),B))) <-> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A),B)))) # label(fact_2024_add__le__cancel__left) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3197 (all M all N_1 all K_2 (one_one_int = hAPP_int_int(legacy_zgcd(N_1),K_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N_1))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M))))) # label(fact_4515_zrelprime__zdvd__zmult) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3198 (all A_207 A_207 = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_207),zero_zero_rat)) # label(fact_635_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3199 (all B_73 all A_84 all C_36 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_36)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_73),A_84)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_73),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_84),C_36)))))) # label(fact_2158_add__increasing2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3200 (all P all Q all R hAPP_r1134773055l_bool(P,hAPP_int_real(Q,R)) = hAPP_i688218208l_bool(hAPP_f801032407l_bool(hAPP_f499530369l_bool(cOMBB_1317819104ol_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__H) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3201 (all A_3 all B_2 (is_int(B_2) & is_int(A_3) -> (hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2) != zero_zero_int -> (exists A_4 exists B_3 (is_int(A_4) & is_int(B_3) & hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_4),B_3) = one_one_int & hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_3),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)) = B_2 & A_3 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_4),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2))))))) # label(fact_5117_gcd__coprime__exists__int) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3202 (all F all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)) -> ((all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),B)) -> (exists Y (hBOOL(hAPP_real_bool(deriv_real(F,X),Y)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),zero_zero_real)))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,B)),hAPP_real_real(F,A)))))) # label(fact_4003_DERIV__neg__imp__decreasing) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3203 (all K_1 all L (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K_1)),hAPP_int_int(bit1,L))))) # label(fact_109_rel__simps_I17_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3204 (all K_2 all L_2 all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),L_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),L_2)))))) # label(fact_1776_add__le__mono) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3205 (all B_125 all A_149 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_149)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_125)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_149),B_125)))))) # label(fact_1242_mult__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3206 (all Y_1 all X_1 (-hBOOL(hAPP_int_bool(even_odd_even_int,X_1)) -> (hBOOL(hAPP_int_bool(even_odd_even_int,Y_1)) -> -hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Y_1)))))) # label(fact_2890_odd__plus__even) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3207 (all X_1 all Y_1 all K_2 all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_1),Y_1) -> (exists I_2 exists J_1 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),J_1) = Y_1 & hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),I_2) = X_1))))) # label(fact_4480_prime__power__mult) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3208 (all X_15 all Y_11 all Ya_1 hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_15),Ya_1)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Y_11),Ya_1)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_15),Y_11)),Ya_1)) # label(fact_2493_mult__left_Odiff) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3209 (all K_2 all I all J_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),J_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J_2),K_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),K_2))))) # label(fact_67_zle__trans) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3210 (all A (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),A))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A)))) # label(fact_2199_zero__less__double__add__iff__zero__less__single__add) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3211 (exists K_4 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),K_4)) & (all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(re,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X)),K_4)))))) # label(fact_4139_Re_Opos__bounded) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3212 (all B_119 all A_142 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_142),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_119),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_142),B_119)))))) # label(fact_1278_mult__neg__neg) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3213 (all A_208 (is_int(A_208) -> A_208 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),A_208))) # label(fact_623_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3214 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> hAPP_real_real(arcsin,hAPP_real_real(sin,X_1)) = X_1))) # label(fact_3739_arcsin__sin) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3215 (all P all Q all R hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,P),Q),R) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Nat__Onat_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3216 (all C_80 all A_164 all B_139 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_164),B_139)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_164),C_80)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_164),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_139),C_80)))))) # label(fact_1145_dvd__add) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3217 (all A_191 all N_42 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_191),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_42))))) # label(fact_859_zero__le__even__power_H) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3218 (all L all U hAPP_n1699378549t_bool(ord_at4362885an_nat(L),hAPP_nat_nat(suc,U)) = hAPP_n1699378549t_bool(ord_at238088361st_nat(L),U)) # label(fact_5074_atLeastLessThanSuc__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3219 (all Z_2 all X_2 all X_3 (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),X_3)) -> ((all X (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X),X_3)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Z_2)))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),comple124823625p_real(X_3)))))) # label(fact_4639_SupInf_OSup__upper) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3220 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),A))))) # label(fact_4721_one__less__Fract__iff) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3221 (all B_138 all A_162 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_138)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_162),zero_zero_rat)) | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_162)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_138),zero_zero_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_162),B_138)),zero_zero_rat)))) # label(fact_1178_split__mult__neg__le) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3222 (all X_43 all Y_35 all Q_10 hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,X_43),Y_35)),Q_10) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_43),Q_10)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_35),Q_10))) # label(fact_341_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3223 (all K_1 all Xa_1 hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,sin),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,plus_plus_real),K_1)),Xa_1),hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Xa_1),K_1))))) # label(fact_4742_DERIV__sin__add) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3224 (all C_44 all D_8 all A_101 all B_84 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_101),B_84)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_44),D_8)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_101),C_44)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_84),D_8)))))) # label(fact_2007_add__mono) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3225 (all N_1 hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),pi)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),N_1)) # label(fact_3714_cos__npi) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3226 (all A_189 all N_40 all N_39 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_40),N_39)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,one_on1645066479umeral),A_189)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_189),N_40)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_189),N_39)))))) # label(fact_869_power__increasing) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3227 (all N_1 one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,N_1),one_one_int)),N_1)) # label(fact_4973_coprime__minus__one__int) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3228 (all B_90 all A_111 all C_50 (hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_50),A_111) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_90),A_111) -> B_90 = C_50)) # label(fact_1914_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3229 (all D_3 all A_3 all B_2 (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)) -> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),B_2) = one_one_int)) # label(fact_4998_coprime__rmult__int) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3230 (all C_1 all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),C_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),C_1))))) # label(fact_1606_divides__mul__r) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3231 (all X_1 all Y_1 ((Y_1 = zero_zero_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Y_1) = X_1) & (Y_1 != zero_zero_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y_1),hAPP_nat_nat(div_mod_nat(X_1),Y_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Y_1)))) # label(fact_4359_gcd__nat_Osimps) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3232 (all N all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),Ma)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,N)),one_one_real)),hAPP_nat_real(real_nat,Ma))))) # label(fact_3684_nat__less__real__le) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3233 (all W (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W))) -> hAPP_int_int(div_mod_int(one_one_int),hAPP_int_int(number_number_of_int,W)) = hAPP_P1175774780nt_int(product_snd_int_int,posDivAlg(one_one_int,hAPP_int_int(number_number_of_int,W))))) # label(fact_4235_mod__pos__pos__1__number__of) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3234 (all A_173 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,zero_z126310315umeral),A_173)) -> A_173 = zero_z126310315umeral)) # label(fact_1079_dvd__0__left) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3235 (all X_1 all Y_1 hAPP_real_real(hAPP_r1250527377l_real(hAPP_b646336293l_real(if_real,fFalse),X_1),Y_1) = Y_1) # label(help_If_2_1_If_000tc__RealDef__Oreal_T) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3236 (all X_2 all Y_2 (hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_2),Y_2))) <-> hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2)) | hBOOL(hAPP_nat_bool(even_odd_even_nat,Y_2)))) # label(fact_3760_even__product__nat) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3237 (all X_1 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_1)),hAPP_real_real(cos,X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X_1)),hAPP_real_real(sin,X_1))) = one_one_real) # label(fact_3679_sin__cos__squared__add3) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3238 (all R_3 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_3)),ii) = hAPP_real_complex(hAPP_r265291036omplex(complex,zero_zero_real),R_3)) # label(fact_3994_complex__of__real__i) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3239 (all M all N_1 all K_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(div_mod_nat(M),N_1)),K_2) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),K_2))) # label(fact_2510_mod__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3240 (all A_1 all B_1 all Ma (hBOOL(hAPP_f1469637905l_bool(hAPP_P1681069257l_bool(member308914164t_bool,hAPP_f538843494t_bool(hAPP_f1418982953t_bool(produc713279741t_bool,A_1),B_1)),bijR_int_int(zcongm(Ma)))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(big_co1548731110nt_int(cOMBI_int,A_1)),big_co1548731110nt_int(cOMBI_int,B_1)),Ma)))) # label(fact_5154_bijzcong__zcong__prod) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3241 (all Z_1 (is_int(Z_1) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,one_one_int),Z_1) = Z_1)) # label(fact_362_zmult__1) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3242 (all X_8 all Q_6 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_8),hAPP_nat_nat(suc,Q_6)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_8),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_8),Q_6))) # label(fact_3015_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3243 (all X_1 all Y_1 hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,Y_1)),hAPP_real_real(sin,X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,Y_1)),hAPP_real_real(cos,X_1)))) # label(fact_3665_sin__diff2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3244 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N_1))))) # label(fact_3063_le__imp__less__Suc) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3245 (all C_79 all D_15 all A_160 all B_136 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_160),B_136)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_79),D_15)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_136)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_79)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_160),C_79)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_136),D_15)))))))) # label(fact_1186_mult__mono) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3246 (all N all X_2 exists T (hAPP_real_real(exp_real,X_2) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(power_power_real,X_2))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(exp_real,T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),N))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_2))))) # label(fact_4925_Maclaurin__exp__le) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3247 (all M all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,M),N_1))) # label(fact_5190_max__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3248 (all X_2 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(number_number_of_rat,X_2)),zero_zero_rat)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),pls)))) # label(fact_761_less__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3249 (all D2 all D1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D2)) -> (exists E (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),E)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,E),D1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,E),D2))))))) # label(fact_3294_real__lbound__gt__zero) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3250 (all B_117 all A_140 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_117),A_140))) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_140)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_117))))) # label(fact_1295_zero__less__mult__pos2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3251 (all P_5 all N (hBOOL(hAPP_nat_bool(prime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_5),N))) <-> hBOOL(hAPP_nat_bool(prime,P_5)) & N = one_one_nat)) # label(fact_4175_prime__exp) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3252 (all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> N_1 = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),one_one_nat)))) # label(fact_3183_Suc__pred_H) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3253 (all X_2 (hAPP_real_real(cos,X_2) != zero_zero_real -> hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,inverse_divide_real),sin)),cos),X_2),hAPP_real_real(inverse_inverse_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X_2)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_4799_lemma__DERIV__tan) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3254 (all A_236 all B_165 all C_96 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_236),C_96)),B_165) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_236),B_165)),C_96)) # label(fact_213_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3255 (all X_40 all Y_32 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_40),X_40)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_32),Y_32))))) # label(fact_729_sum__squares__ge__zero) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3256 (all C_77 all B_134 all A_158 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_134),A_158)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_77),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_77),A_158)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_77),B_134)))))) # label(fact_1195_mult__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3257 (all D all Sa exists Z all X (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X),Z)) -> (-hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X),Sa))) <-> -hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X),Sa)))))) # label(fact_2266_minf_I10_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3258 (all B_129 all A_153 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_153),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_129),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_153),B_129)))))) # label(fact_1219_mult__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3259 (all A all C all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A),B)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A),C)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B),C))))) # label(fact_2032_add__le__cancel__right) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3260 (all A_75 zero_zero_nat = hAPP_nat_nat(div_mod_nat(A_75),A_75)) # label(fact_2303_mod__self) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3261 (all A_3 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),hAPP_int_int(div_mod_int(A_3),B_2))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(div_mod_int(A_3),B_2)),zero_zero_int)))) # label(fact_2449_neg__mod__conj) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3262 (all A_133 hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_133),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_133),one_on1684967323de_int)))) # label(fact_1372_less__add__one) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3263 (all A_179 all B_151 (hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_179),B_151) = zero_zero_complex -> zero_zero_complex = A_179 | B_151 = zero_zero_complex)) # label(fact_1022_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3264 (all P all Q all R hAPP_real_bool(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,P),Q),R) = hAPP_bool_bool(hAPP_r1938957037l_bool(P,R),hAPP_real_bool(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__HOL__Obool_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3265 (all P all Q all R hAPP_nat_fun_rat_rat(hAPP_f1830834240at_rat(hAPP_f1073644437at_rat(cOMBB_1926947at_nat,P),Q),R) = hAPP_rat_fun_rat_rat(P,hAPP_nat_rat(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Rat__Orat_000tc__fun_Itc__Rat__Orat_Mtc__Rat__Orat_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3266 (all A_9 all N_15 all B_8 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_9),hAPP_nat_nat(suc,N_15))),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_8),hAPP_nat_nat(suc,N_15)))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_8)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_9),B_8))))) # label(fact_3107_power__le__imp__le__base) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3267 (all M_11 -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,M_11)),zero_zero_nat))) # label(fact_2922_of__nat__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3268 (all B_113 all A_136 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_136)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_113)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_136),B_113)))))) # label(fact_1318_mult__pos__pos) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3269 (all K_3 all K_1 all F ((all N_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,N_2)),K_3)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,K_1),hAPP_nat_real(F,N_2))))) -> hBOOL(bseq_real(F)))) # label(fact_4947_BseqI2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3270 (all C_69 all A_144 all B_121 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_144),B_121)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_69)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_69),A_144)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_69),B_121)))))) # label(fact_1266_mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3271 (all A_47 zero_zero_nat = hAPP_nat_nat(div_mod_nat(A_47),one_one_nat)) # label(fact_2416_mod__by__1) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3272 (all X_2 all F all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)) -> ((all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),B)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X)) -> hBOOL(hAPP_real_bool(isCont_real_real(F),X)))) -> ((all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),B)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),X)) -> hBOOL(hAPP_real_bool(deriv_real(F,X),zero_zero_real)))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),B)) -> hAPP_real_real(F,A) = hAPP_real_real(F,X_2))))))) # label(fact_5176_DERIV__isconst2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3273 (all A all B all C (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A),B)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),A)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),B)))))) # label(fact_1403_mult__le__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3274 (all A all B (hAPP_C498520661umeral(div_mo1740067990umeral(B),A) = zero_z126310315umeral <-> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A),B)))) # label(fact_2431_dvd__eq__mod__eq__0) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3275 (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(ln,X_2)),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),one_one_real))))) # label(fact_3441_ln__less__zero__iff) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3276 (all X_2 all Y_2 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),Y_2)),zEven)) <-> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven)) <-> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven))))) # label(fact_2900_even__diff) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3277 (all A_62 all B_54 all C_25 hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_62),hAPP_Q1762011733de_int(div_mo231679042de_int(B_54),C_25))),C_25) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_62),B_54)),C_25)) # label(fact_2353_mod__add__right__eq) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3278 (all Z_1 all X_1 all Y_1 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),Z_1)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_1),Y_1)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_1),X_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Z_1))))) # label(fact_1717_dvd_Oless__trans) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3279 (all W_1 all X_2 (X_2 = hAPP_int_nat(number_number_of_nat,W_1) <-> hAPP_int_nat(number_number_of_nat,W_1) = X_2)) # label(fact_251_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3280 (all N_1 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N_1)),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_1)))) -> N_1 = hAPP_int_nat(nat,archim1246769320r_real(X_1))))) # label(fact_4034_floor__eq4) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3281 (all N_1 -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),zero_zero_nat))) # label(fact_1740_not__less0) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3282 (all P all Q all R hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,P),Q),R) = hAPP_f448129468l_bool(hAPP_i2112223885l_bool(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3283 (all N_1 -hBOOL(hAPP_int_bool(nat_neg,hAPP_nat_int(semiri1621563631at_int,N_1)))) # label(fact_3272_not__neg__int) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3284 (all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K_1)),pls)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),pls)))) # label(fact_274_rel__simps_I27_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3285 (all C_81 all A_167 all B_142 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_167),B_142)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_167),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_142),C_81))))) # label(fact_1121_dvd__mult2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3286 (all W_1 all Y_2 all X_2 all Z_2 (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,W_1),Y_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_2),Z_2)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,W_1),Z_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_2),Y_2)) <-> W_1 = X_2 | Y_2 = Z_2)) # label(fact_298_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3287 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X_1))),X_1))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3473_abs__ln__one__plus__x__minus__x__bound__nonneg) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3288 (all P all Q all R hAPP_f429384717t_bool(P,hAPP_i627165055t_real(Q,R)) = hAPP_i1948725293t_bool(hAPP_f351634443t_bool(hAPP_f1576199059t_bool(cOMBB_95956018ol_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__RealDef__Oreal_J_000tc__fun_) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3289 (all A_165 all B_140 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_165),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_165),B_140)))) # label(fact_1139_dvd__triv__left) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3290 (all Y_2 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(hAPP_n546711566l_real(root,N),Y_2)))))) # label(fact_3926_real__root__gt__1__iff) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3291 (all X_1 (is_int(X_1) -> X_1 = hAPP_int_int(succ,hAPP_int_int(pred,X_1)))) # label(fact_4038_succ__pred) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3292 (all A_3 all B_2 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),A_3)),B_2))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_7_zadd__power2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3293 (all A_90 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_90),one_one_rat) = A_90) # label(fact_2113_mult__1__right) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3294 (all X_2 all N all Y_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_2),hAPP_nat_nat(suc,N))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_2),hAPP_nat_nat(suc,N)))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),Y_2)))) # label(fact_3091_exp__mono__le) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3295 (all X_1 hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),pi))) = hAPP_real_real(cos,X_1)) # label(fact_3582_cos__periodic) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3296 (all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),one_one_real)) -> (exists X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X)) & Y_1 = hAPP_real_real(sin,X) & (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),Y)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) & hAPP_real_real(sin,Y) = Y_1 -> Y = X)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))))) # label(fact_3746_sin__total) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3297 (all R all S_4 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R),hAPP_real_real(inverse_inverse_real,S_4)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,R),S_4)) # label(fact_3829_real__divide__def) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3298 (all Ta all D all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),D_1)) -> (all X all K (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D_1))),Ta))) <-> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Ta))))))) # label(fact_1820_inf__period_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3299 (all N_27 all Y_8 all X_12 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,Y_8),X_12)) -> hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_12),N_27)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_8),N_27)) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,X_12),Y_8)),N_27))) # label(fact_2617_div__power) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3300 (all B all A all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int))) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(inv(P_5),B)),hAPP_i1948725293t_bool(wset(A),P_5))) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(wset(A),P_5)))))))))) # label(fact_2230_wset__inv__mem__mem) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3301 (all X hAPP_n1699378549t_bool(ord_less_nat,X) = hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,X))) # label(fact_4740_less__eq__Suc__le__raw) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3302 (all A_3 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(div_mod_int(A_3),B_2))))) # label(fact_2453_pos__mod__sign) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3303 (all X_49 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_49),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_49),X_49)) # label(fact_33_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3304 (all X_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(cos,X_1)),one_one_real))) # label(fact_3561_cos__le__one) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3305 (all C_83 all D_16 all A_169 all B_144 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_169),B_144)) -> (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,C_83),D_16)) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_169),C_83)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_144),D_16)))))) # label(fact_1108_mult__dvd__mono) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3306 (all K_2 hAPP_int_real(number267125858f_real,K_2) = ratreal(hAPP_int_rat(number_number_of_rat,K_2))) # label(fact_4696_Ratreal__number__collapse_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3307 (all K_1 hAPP_n1699378549t_bool(ord_atMost_nat,K_1) = hAPP_n1699378549t_bool(ord_lessThan_nat,hAPP_nat_nat(suc,K_1))) # label(fact_5186_lessThan__Suc__atMost) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3308 (all P_1 all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) -> (all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),hAPP_nat_nat(suc,N_2)))),hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_2))))))) # label(fact_4245_Bolzano__bisect__Suc__le__snd) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3309 (all X_1 all Y_1 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(real_int,X_1)),hAPP_int_real(real_int,Y_1)) = hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_1))) # label(fact_4395_real__of__int__mult) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3310 (all U all F ((all N_2 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_2),hAPP_nat_nat(F,N_2)))) -> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,ord_less_eq_nat),F)),U)))))) # label(fact_4894_finite__less__ub) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3311 (all X_2 hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_eq_nat),X_2))) = hAPP_nat_nat(suc,X_2)) # label(fact_4745_card__bdd__nat__set__le) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3312 (all Ma all K_1 all N (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,K_1),N) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),Ma)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ma),N)))))) # label(fact_4964_coprime__dvd__mult__iff__int) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3313 (all V_4 ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_4))) -> zero_zero_rat = hAPP_nat_rat(semiri151668891at_rat,hAPP_int_nat(number_number_of_nat,V_4))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_4))) -> hAPP_int_rat(number_number_of_rat,V_4) = hAPP_nat_rat(semiri151668891at_rat,hAPP_int_nat(number_number_of_nat,V_4))))) # label(fact_3218_of__nat__number__of__lemma) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3314 (all R_3 all X_1 all Y_1 hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),X_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),Y_1)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_3)),hAPP_real_complex(hAPP_r265291036omplex(complex,X_1),Y_1))) # label(fact_3990_complex__of__real__mult__Complex) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3315 (all A_64 all B_56 hAPP_Q1762011733de_int(div_mo231679042de_int(A_64),B_56) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_64),B_56)),B_56)) # label(fact_2345_mod__add__self2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3316 (all W hAPP_int_nat(number_number_of_nat,W) = hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,W))) # label(fact_2729_nat__number__of) # label(axiom) # label(non_clause). [assumption]. 5.65/5.68 3317 (all Ma all N (hAPP_nat_nat(suc,zero_zero_nat) = Ma & hAPP_nat_nat(suc,zero_zero_nat) = N <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),N) = hAPP_nat_nat(suc,zero_zero_nat))) # label(fact_3055_mult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3318 (all B_93 all A_114 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_114)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_93)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_114),B_93)))))) # label(fact_1885_add__nonneg__pos) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3319 (all K_2 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,K_2)),zero_zero_int))) # label(fact_2986_int__less__0__conv) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3320 (all N (N = one_one_nat <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,N),N)))) # label(fact_4012_coprime__refl) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3321 (all X_2 hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),sin_coeff)),hAPP_r474017924t_real(power_power_real,X_2))))) # label(fact_4830_summable__sin) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3322 (all N_16 all M_5 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_16),M_5)) -> hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_nat_rat(semiri151668891at_rat,M_5)),hAPP_nat_rat(semiri151668891at_rat,N_16)) = hAPP_nat_rat(semiri151668891at_rat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_5),N_16)))) # label(fact_3101_of__nat__diff) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3323 (all D all Sa exists Z all X (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,Z),X)) -> (-hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,D),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X),Sa))) <-> -hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,D),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X),Sa)))))) # label(fact_2257_pinf_I10_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3324 (all I all K_2 all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_2)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),I)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_2)),I))) # label(fact_1669_diff__add__assoc2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3325 (all A_89 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,one_one_real),A_89) = A_89) # label(fact_2116_mult__1) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3326 (all N_1 hAPP_nat_nat(suc,hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_3199_add__2__eq__Suc_H) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3327 (all K_2 all N_1 all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,K_2),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,K_2),M)))))) # label(fact_2733_div__le__mono2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3328 (all X_2 hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),sin)),sin),X_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(cos,X_2))),hAPP_real_real(sin,X_2))))) # label(fact_4789_DERIV__sin__sin__mult2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3329 (all N all V (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))) & (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N)),hAPP_int_nat(number_number_of_nat,V))))) # label(fact_4784_le__Suc__number__of) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3330 (all K_1 (is_int(K_1) -> (min = K_1 <-> hAPP_int_int(bit1,K_1) = min))) # label(fact_924_rel__simps_I47_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3331 (all A_97 all C_40 all B_80 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_97),C_40)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_80),C_40))) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_97),B_80)))) # label(fact_2042_add__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3332 (all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arcsin,Y_1)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3508_arcsin__ubound) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3333 (all K_6 all M_16 all N_29 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,K_6),hAPP_C498520661umeral(div_mo1740067990umeral(M_16),N_29))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,K_6),N_29)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,K_6),M_16))))) # label(fact_2378_dvd__mod__imp__dvd) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3334 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),zero_zero_int)) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),A_3)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_2),one_one_int)),A_3))) # label(fact_2533_neg__zdiv__mult__2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3335 (all N_1 all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N_1),M))))) # label(fact_1797_nat__dvd__not__less) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3336 (all P all Q all R hAPP_rat_bool(P,hAPP_P1853363688nt_rat(Q,R)) = hAPP_P603027463t_bool(hAPP_f1084724746t_bool(hAPP_f374502857t_bool(cOMBB_569260360nt_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Rat__Orat_000tc__HOL__Obool_000tc__prod_Itc__Int__Oi) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3337 (all A all B all C (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B),A)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),A)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B)))))) # label(fact_1405_mult__le__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3338 (all M all N_1 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N_1)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1))) # label(fact_3005_zmult__int) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3339 (all P (hBOOL(hAPP_bool_bool(fNot,P)) | hBOOL(P))) # label(help_fNot_2_1_U) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3340 (all A_124 all C_58 all B_103 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_124),C_58)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_103),C_58))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_58)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_124),B_103))))) # label(fact_1458_mult__right__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3341 (all K_1 all Ma all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)))) # label(fact_1802_mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3342 (all B_150 all A_178 (A_178 != zero_zero_complex -> (B_150 != zero_zero_complex -> hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_178),B_150) != zero_zero_complex))) # label(fact_1029_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3343 (all N ((exists M_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),M_1) = N) <-> hBOOL(hAPP_nat_bool(even_odd_even_nat,N)))) # label(fact_3784_even__mult__two__ex) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3344 (all Z_1 (is_int(Z_1) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),zero_zero_int) = Z_1)) # label(fact_647_zadd__0__right) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3345 (all X_32 one_one_real = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_32),zero_zero_nat)) # label(fact_950_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3346 (all Ma all N (hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),N))) <-> (hBOOL(hAPP_nat_bool(even_odd_even_nat,N)) <-> hBOOL(hAPP_nat_bool(even_odd_even_nat,Ma))))) # label(fact_3755_even__add) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3347 (all X_2 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,X_2)),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Y_2))))) # label(fact_3492_real__0__less__add__iff) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3348 (all Z_12 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_12) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Z_12),Z_12)) # label(fact_490_semiring__mult__2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3349 (all N_1 all D_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),D_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),N_1)) -> hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N_1),D_3)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,N_1)),hAPP_nat_real(real_nat,D_3))))) # label(fact_3695_real__of__nat__div) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3350 (all N_32 all M_18 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),M_18)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),N_32)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,M_18),N_32)))))) # label(fact_1364_less__1__mult) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3351 (all V_2 all W hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_2),hAPP_int_int(uminus_uminus_int,W))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,W))) # label(fact_3542_minus__numeral__code_I6_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3352 (all N_11 all A_6 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_6)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_6),one_one_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_6),hAPP_nat_nat(suc,N_11))),one_one_nat))))) # label(fact_3166_power__Suc__less__one) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3353 (all C_46 all A_103 all B_86 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_46),A_103)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_46),B_86))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_103),B_86)))) # label(fact_1997_add__le__imp__le__left) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3354 (all Ma all N all K_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N))))) # label(fact_1845_nat__mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3355 (all B_70 all A_81 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_81),zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_70),zero_zero_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_81),B_70)),zero_zero_nat))))) # label(fact_2186_add__neg__neg) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3356 (all X_2 hAPP_real_real(sin,X_2) = hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),sin_coeff)),hAPP_r474017924t_real(power_power_real,X_2)))) # label(fact_4818_sin__def) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3357 (all A all A_1 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma)) -> (hBOOL(hAPP_int_bool(is_RRset(A_1),Ma)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),A_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),hAPP_int_int(rRset2norRR(A_1,Ma),A)),Ma)))))) # label(fact_4592_RRset2norRR__correct1) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3358 (all N_67 all A_218 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_218)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_218),N_67)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_218),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_218),N_67)))))) # label(fact_519_power__less__power__Suc) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3359 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A),B)))))) # label(fact_2633_pos__imp__zdiv__nonneg__iff) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3360 (all A_54 all B_46 all C_18 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_54),B_46)),C_18) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(div_mod_int(A_54),C_18)),B_46)),C_18)) # label(fact_2375_mod__diff__left__eq) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3361 (all K_6 all M_16 all N_29 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_6),hAPP_int_int(div_mod_int(M_16),N_29))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_6),N_29)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_6),M_16))))) # label(fact_2380_dvd__mod__imp__dvd) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3362 (all W (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,W)),zero_zero_int)) -> hAPP_P1175774780nt_int(product_snd_int_int,hAPP_P1975530577nt_int(negateSnd,negDivAlg(hAPP_int_int(uminus_uminus_int,one_one_int),hAPP_int_int(uminus_uminus_int,hAPP_int_int(number_number_of_int,W))))) = hAPP_int_int(div_mod_int(one_one_int),hAPP_int_int(number_number_of_int,W)))) # label(fact_4231_mod__pos__neg__1__number__of) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3363 (all A_175 all E_3 all B_149 all C_87 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_175),E_3)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_149),E_3)),C_87)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_175),B_149)),E_3)),C_87)) # label(fact_1066_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3364 (all X_1 ((X_1 = zero_zero_real -> zero_zero_real = hAPP_real_real(sgn_sgn_real,X_1)) & (X_1 != zero_zero_real -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> one_one_real = hAPP_real_real(sgn_sgn_real,X_1)) & (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> hAPP_real_real(uminus_uminus_real,one_one_real) = hAPP_real_real(sgn_sgn_real,X_1))))) # label(fact_3575_real__sgn__def) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3365 (all X_1 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sin,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1))) # label(fact_3736_cos__double) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3366 (all N_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N_1)) -> zero_zero_int != hAPP_int_int(fact_fact_int,N_1))) # label(fact_4277_fact__nonzero__int) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3367 (all A all B all C (C = B <-> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A),C) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A),B))) # label(fact_1943_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3368 (all Y_1 all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Y_1)))))) # label(fact_1391_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3369 (all A_234 all C_94 all D_19 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_234),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_94),D_19)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_234),C_94)),D_19)) # label(fact_231_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3370 (all N all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N),Ma)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,Ma)),one_one_real))))) # label(fact_4447_int__le__real__less) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3371 (all X_1 all Y_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),X_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_1),Y_1)))))) # label(fact_3335_le__real__sqrt__sumsq) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3372 (all Ma all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)))) # label(fact_2992_zless__int) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3373 (all Lx_2 all Ly all Rx_2 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_2),Ly)),Rx_2) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_2),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Ly),Rx_2))) # label(fact_170_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3374 (all U hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(ord_at641636577an_int(zero_zero_int),U)) = hAPP_int_nat(nat,U)) # label(fact_4918_card__atLeastZeroLessThan__int) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3375 (all K_2 all M all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_1))) # label(fact_1656_diff__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3376 (all X_1 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(sin,X_1))),hAPP_real_real(cos,X_1)) = hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1))) # label(fact_3709_sin__double) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3377 (all W_1 (hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W_1)))) <-> hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,W_1))))) # label(fact_3277_neg__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3378 (all X_1 all Y_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Y_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_1),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),Y_1))),X_1)))) # label(fact_2687_zdiv__leq__prop) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3379 (all I_1 all M_2 (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,zero_zero_nat),M_2)) -> hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),M_2))),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),hAPP_nat_nat(suc,I_1))))) = hAPP_nat_nat(suc,hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,hAPP_f66927821l_bool(hAPP_f2037783381l_bool(cOMBB_1146692694ol_nat,member_nat),suc)),M_2))),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),I_1))))))) # label(fact_4769_card__less__Suc) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3380 (all A_3 all B_2 all P_3 all Q_2 (quotient_of(hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2) -> hAPP_int_rat(hAPP_int_fun_int_rat(fract,P_3),Q_2) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2))) # label(fact_4668_quotient__of__eq) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3381 (all N_1 all M hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,N_1))),hAPP_nat_int(semiri1621563631at_int,M)))) # label(fact_3509_negative__zle) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3382 (all X_36 all Y_28 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_36),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_28),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),Y_28)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_36),Y_28))))) # label(fact_843_power2__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3383 (all N_1 -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_1)),N_1))) # label(fact_2877_Suc__n__not__le__n) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3384 (all X_1 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sin,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = one_one_real) # label(fact_3728_sin__cos__squared__add2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3385 (all N_35 all A_182 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_182)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_35)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_182),N_35)))))) # label(fact_975_one__less__power) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3386 (all Y_2 all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),X_2)) -> comple124823625p_real(hAPP_r1134773055l_bool(ord_at1496968948n_real(Y_2),X_2)) = X_2)) # label(fact_4916_Sup__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3387 (all B_2 all A_3 all X_1 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),A_3)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),X_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),X_1)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2))))) # label(fact_4313_gcd__semilattice__nat_Oless__infI1) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3388 (all V_2 ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls)) -> hAPP_int_nat(number_number_of_nat,hAPP_int_int(succ,V_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_int_nat(number_number_of_nat,V_2))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls)) -> one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_int_nat(number_number_of_nat,V_2))))) # label(fact_3302_nat__1__add__number__of) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3389 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(number_number_of_nat,pls)),N_1)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),N_1))) # label(fact_3200_Suc__diff__eq__diff__pred) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3390 (all Y_22 all X_28 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_28)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_22)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_22),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_22),X_28)),X_28)))))) # label(fact_1468_mult__left__le__one__le) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3391 (all X_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sin,X_1)),one_one_real))) # label(fact_3643_sin__le__one) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3392 (all P_3 all Q_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Q_2)) -> (hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,P_3),Q_2) = one_one_int -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2) = normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2))))) # label(fact_5021_normalize__stable) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3393 (all B_114 all A_137 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_137)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_114),zero_z891286103de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_137),B_114)),zero_z891286103de_int))))) # label(fact_1312_mult__pos__neg) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3394 (all C_39 all D_7 all A_96 all B_79 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_96),B_79)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_39),D_7)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_96),C_39)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_79),D_7)))))) # label(fact_2051_add__strict__mono) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3395 (all A_175 all E_3 all B_149 all C_87 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_175),B_149)),E_3)),C_87) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_175),E_3)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_149),E_3)),C_87))) # label(fact_1069_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3396 (all Ta all A_1 all D all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),D_1)) -> (all X (is_int(X) -> ((all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A_1)) -> X != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa))))))) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Ta))) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),D_1)),Ta))))))))) # label(fact_5101_aset_I10_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3397 (all Z_1 hAPP_complex_complex(cnj,hAPP_complex_complex(cnj,Z_1)) = Z_1) # label(fact_4071_complex__cnj__cnj) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3398 (all N_1 hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),N_1)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3722_fact__lemma) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3399 (all X_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)) = hAPP_real_real(hAPP_n546711566l_real(root,N_1),hAPP_real_real(inverse_inverse_real,X_1)))) # label(fact_3914_real__root__inverse) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3400 (all C_73 all A_154 all B_130 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_154),B_130)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_73)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_154),C_73)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_130),C_73)))))) # label(fact_1214_mult__right__mono) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3401 (all X_1 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sin,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_3730_cos__squared__eq) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3402 (all Z_11 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_11) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_11),Z_11)) # label(fact_492_mult__2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3403 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(tan,X_1)))))) # label(fact_3613_tan__gt__zero) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3404 (all P all Q all R hAPP_real_real(hAPP_f1122023986l_real(hAPP_f1786887945l_real(cOMBB_1896127834l_real,P),Q),R) = hAPP_f352196356l_real(P,hAPP_r474017924t_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_000tc__Real) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3405 (all A_238 all B_167 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_238),B_167) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_167),A_238)) # label(fact_187_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3406 (all R_3 ((hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),zero_zero_real)) -> hAPP_real_real(uminus_uminus_real,R_3) = hAPP_real_real(abs_abs_real,R_3)) & (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),zero_zero_real)) -> R_3 = hAPP_real_real(abs_abs_real,R_3)))) # label(fact_3493_real__abs__def) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3407 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,X_1)),one_one_real)) -> (exists Z (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),Z)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) & X_1 = hAPP_real_real(tan,Z))))) # label(fact_3624_tan__total__pi4) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3408 (all A_174 all B_148 all C_86 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_174),C_86)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_148),C_86)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_174),B_148)),C_86)) # label(fact_1073_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3409 (all A_3 all B_2 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2))) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),zero_zero_int)) -> negDivAlg(A_3,B_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2))) & (-(hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2))) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),zero_zero_int))) -> hAPP_P1975530577nt_int(adjust(B_2),negDivAlg(A_3,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_2))) = negDivAlg(A_3,B_2)))) # label(fact_2495_negDivAlg_Osimps) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3410 (all A_229 all B_162 all C_91 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_229),B_162)),C_91) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_229),C_91)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_162),C_91))) # label(fact_314_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3411 (all P (is_bool(P) -> fTrue = P | P = fFalse)) # label(help_If_3_1_If_000tc__Int__Oint_T) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3412 (all A_3 all B_2 hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) = archim791455193or_rat(hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2))) # label(fact_4682_floor__Fract) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3413 (all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,min)),B_2) = hAPP_int_int(number_number_of_int,min))) # label(fact_2689_div__eq__minus1) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3414 (all A_10 all N_17 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_10),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_10),N_17)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_10),hAPP_nat_nat(suc,N_17))) # label(fact_3039_power__Suc) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3415 (all X_1 all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sqrt,X_1)),hAPP_real_real(sqrt,Y_1))))) # label(fact_3313_real__sqrt__le__mono) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3416 (all Z_2 all Z_3 ((hBOOL(hAPP_int_bool(nat_neg,Z_3)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat,Z_2)),hAPP_int_nat(nat,Z_3)) = hAPP_int_nat(nat,Z_2)) & (-hBOOL(hAPP_int_bool(nat_neg,Z_3)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat,Z_2)),hAPP_int_nat(nat,Z_3)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,hAPP_int_bool(nat_neg,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_2),Z_3))),zero_zero_nat),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_2),Z_3)))))) # label(fact_4751_diff__nat__eq__if) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3417 (all B_5 all Q_8 all R_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_5),Q_8)),R_5))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_5),B_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_5)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Q_8)))))) # label(fact_1499_q__pos__lemma) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3418 (all N_44 one_one_int = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,one_one_int),N_44)) # label(fact_825_power__one) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3419 (all Z_2 all W_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),W_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat,W_1)),hAPP_int_nat(nat,Z_2)))))) # label(fact_2752_nat__less__eq__zless) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3420 (all N_28 all K_5 all M_15 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_5),M_15)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_5),N_28)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_5),hAPP_int_int(div_mod_int(M_15),N_28)))))) # label(fact_2384_dvd__mod) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3421 (all W_5 ((zero_zero_nat != hAPP_int_nat(number_number_of_nat,W_5) -> zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),hAPP_int_nat(number_number_of_nat,W_5))) & (zero_zero_nat = hAPP_int_nat(number_number_of_nat,W_5) -> one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),hAPP_int_nat(number_number_of_nat,W_5))))) # label(fact_1480_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3422 (all X_2 all Y_2 all B (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),B)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B),X_2)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B),Y_2))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),Y_2))))) # label(fact_539_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3423 (all C_78 all D_14 all A_159 all B_135 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_159),B_135)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_78),D_14)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_159)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_78)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_159),C_78)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_135),D_14)))))))) # label(fact_1189_mult__mono_H) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3424 (all Y_2 all X_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2)) -> ((hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven)) <-> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven))) -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_nat(nat,X_2)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_nat(nat,Y_2)))))) # label(fact_3241_neg__one__power__parity) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3425 (all N_25 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(semiri132038758t_real,N_25)))) # label(fact_2909_of__nat__0__le__iff) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3426 (all A all B all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(div_mod_int(A),Ma)),hAPP_int_int(div_mod_int(B),Ma)),Ma)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma)))) # label(fact_2400_zcong__zmod) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3427 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natceiling(X_1)),one_one_nat) = natceiling(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),one_one_real)))) # label(fact_3408_natceiling__add__one) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3428 (all V_2 (hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> zero_zero_nat = hAPP_int_nat(number_number_of_nat,V_2))) # label(fact_3281_neg__imp__number__of__eq__0) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3429 (all V_5 all W_3 all C_54 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_5)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(number267125858f_real,W_3)),C_54)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_5),W_3))),C_54)) # label(fact_1582_add__number__of__diff1) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3430 (all N_1 all M ((zero_zero_nat = M -> zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1)) & (zero_zero_nat != M -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1)))) # label(fact_1731_mult__eq__if) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3431 (all B all A (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(d22set,A))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B)))) # label(fact_2272_d22set__g__1) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3432 (all N_1 all M_4 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_1)),M_4)) -> (exists M_1 hAPP_nat_nat(suc,M_1) = M_4))) # label(fact_3269_Suc__le__D) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3433 (all K_2 all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N_1))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),N_1)))) # label(fact_4315_dvd__gcd__D2__nat) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3434 (all P all Q all R hAPP_i418383825t_bool(hAPP_f1222178131t_bool(hAPP_f743790483t_bool(cOMBB_1647910634ol_int,P),Q),R) = hAPP_f737665877t_bool(P,hAPP_i157317923t_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_000tc__fun_) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3435 (all M all K_2 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),M)),K_2))))) # label(fact_1677_le__add__diff) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3436 (all W_5 ((hAPP_int_nat(number_number_of_nat,W_5) = zero_zero_nat -> one_on1684967323de_int = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,zero_z891286103de_int),hAPP_int_nat(number_number_of_nat,W_5))) & (hAPP_int_nat(number_number_of_nat,W_5) != zero_zero_nat -> zero_z891286103de_int = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,zero_z891286103de_int),hAPP_int_nat(number_number_of_nat,W_5))))) # label(fact_1484_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3437 (all M_10 all N_24 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_10),N_24)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(semiri1619134803umeral,M_10)),hAPP_n1108039445umeral(semiri1619134803umeral,N_24))))) # label(fact_2937_less__imp__of__nat__less) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3438 (all A_107 A_107 = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_107),zero_z126310315umeral)) # label(fact_1963_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3439 (all N_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))))) # label(fact_3848_inv__real__of__nat__fact__ge__zero) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3440 (all B all A (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),zOdd)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),zOdd)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),B)),zEven))))) # label(fact_3097_IntNatAux_Oodd__plus__odd) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3441 (all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(real_nat,N))))) # label(fact_3678_real__of__nat__gt__zero__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3442 (all N_1 all M hAPP_i1732201573de_int(quickcheck_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,quickcheck_int_of(N_1)),quickcheck_int_of(M))) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,N_1),M)) # label(fact_4559_times__code__int__def) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3443 (all X_29 all Y_23 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,X_29),Y_23)) -> (X_29 != Y_23 -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,X_29),Y_23))))) # label(fact_1010_order__le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3444 (all Y_1 all X_1 all P_3 (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X_1),P_3)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),Y_1))))) # label(fact_897_Euler_Oaux____1) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3445 (all N_1 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,N_1),P_3)) -> one_one_int = hAPP_int_int(legacy_zgcd(N_1),P_3))))) # label(fact_4523_zless__zprime__imp__zrelprime) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3446 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) -> (N_1 != hAPP_nat_nat(suc,M) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),N_1))))) # label(fact_2863_Suc__lessI) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3447 (all A_234 all C_94 all D_19 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_234),C_94)),D_19) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_234),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_94),D_19))) # label(fact_229_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3448 (all A_88 A_88 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),A_88)) # label(fact_2122_mult__1__left) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3449 (all R_3 all S_1 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,R_3),S_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,R_3),hAPP_real_real(uminus_uminus_real,S_1))) # label(fact_3485_real__diff__def) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3450 (all X_1 all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),pi)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(cos,X_1)),hAPP_real_real(cos,Y_1))))))) # label(fact_3565_cos__monotone__0__pi_H) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3451 (all W_1 all Y_2 all X_2 all Z_2 (is_int(W_1) & is_int(Y_2) & is_int(X_2) & is_int(Z_2) -> (X_2 = W_1 | Z_2 = Y_2 <-> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W_1),Y_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_2),Z_2)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W_1),Z_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_2),Y_2))))) # label(fact_296_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3452 (all X_1 all Y_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_1),Y_1)),hAPP_nat_nat(nat_gcd(X_1),Y_1)) = hAPP_nat_nat(nat_lcm(X_1),Y_1)) # label(fact_4609_Nitpick_Onat__lcm__def) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3453 (all V all V_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),V_1)) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),V_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),V_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_int_nat(number_number_of_nat,V_1))))) # label(fact_734_less__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3454 (all K_2 hAPP_int_real(number267125858f_real,K_2) = ratreal(hAPP_int_rat(number_number_of_rat,K_2))) # label(fact_4695_number__of__real__code) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3455 (all A_42 all B_35 all B_34 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_42),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_35),B_34)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_42),B_35)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_42),B_34))) # label(fact_2479_mult_Oadd__right) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3456 (all A_184 hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_184),zero_z126310315umeral))) # label(fact_912_dvd__0__right) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3457 (all C_66 all D_13 all A_132 all B_111 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_132),B_111)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_66),D_13)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_111)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_66)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_132),C_66)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_111),D_13)))))))) # label(fact_1412_mult__strict__mono) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3458 (all C_74 all A_155 all B_131 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_155),B_131)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_74)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_74),A_155)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_74),B_131)))))) # label(fact_1207_mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3459 (all A_3 all B_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)) # label(fact_4306_gcd__semilattice__nat_Oinf_Oleft__idem) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3460 (all W_6 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),hAPP_int_int(number_number_of_int,W_6))),hAPP_int_int(number_number_of_int,W_6)) = hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W_6))) # label(fact_742_number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3461 (all Y_1 all X_1 (hBOOL(hAPP_int_bool(even_odd_even_int,X_1)) -> (hBOOL(hAPP_int_bool(even_odd_even_int,Y_1)) -> hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Y_1)))))) # label(fact_2892_Parity_Oeven__plus__even) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3462 (all A_227 A_227 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,one_one_real),A_227)) # label(fact_336_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3463 (all X_1 all N_1 hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,N_1)),pi))) = hAPP_real_real(tan,X_1)) # label(fact_3600_tan__periodic__n) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3464 (all C_75 all A_156 all B_132 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_156),B_132)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_75)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_75),A_156)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_75),B_132)))))) # label(fact_1200_comm__mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3465 (all A_135 -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_135),A_135)),zero_zero_real))) # label(fact_1330_not__square__less__zero) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3466 (all N_41 all A_190 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),A_190)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_190),N_41))))) # label(fact_862_one__le__power) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3467 (all C_57 all A_123 all B_102 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_57),A_123)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_57),B_102))) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_57)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_123),B_102))))) # label(fact_1466_mult__left__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3468 (all A_210 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,zero_zero_complex),A_210) = zero_zero_complex) # label(fact_612_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3469 (all A_27 all B_24 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_27),B_24)) -> hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_27),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,B_24),A_27)) = B_24)) # label(fact_2589_dvd__mult__div__cancel) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3470 (all X_2 (is_int(X_2) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zOdd)) -> -(all K (is_int(K) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),K)),one_one_int) != X_2))))) # label(fact_2707_zOddE) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3471 (all A_38 all A_37 all B_30 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_38),B_30)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_37),B_30)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_38),A_37)),B_30)) # label(fact_2490_mult_Odiff__left) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3472 (all X_46 X_46 = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_46),one_one_nat)) # label(fact_76_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3473 (all A_187 all N_37 all N_36 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_37),N_36)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_187)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_187),N_37)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_187),N_36)))))) # label(fact_893_power__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3474 (all N_1 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N_1)),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N_1)),one_one_real))) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N_1),one_one_int) = archim856651990g_real(X_1)))) # label(fact_4473_ceiling__eq) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3475 (all A_106 A_106 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_106),zero_zero_nat)) # label(fact_1972_add__0__right) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3476 (all X_1 all N_5 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_5)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,N_5),X_1)),hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1))))))) # label(fact_3921_real__root__decreasing) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3477 (all N_1 all D_3 (is_int(D_3) -> (D_3 != zero_zero_int -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_3),N_1)) -> hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,N_1)),hAPP_int_real(real_int,D_3)) = hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N_1),D_3)))))) # label(fact_4449_real__of__int__div) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3478 (all A_133 hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_133),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_133),one_on1645066479umeral)))) # label(fact_1371_less__add__one) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3479 (all A_3 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)),B_2)))) # label(fact_5010_gcd__le2__int) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3480 (all A (zero_zero_complex = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) <-> A = zero_zero_complex)) # label(fact_778_zero__eq__power2) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3481 (all X_2 all F (hBOOL(summable_real(F)) -> ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_2))),X_2))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f352196356l_real(suminf_real,F)),X_2))))) # label(fact_4943_suminf__le) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3482 (all N_9 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,N_9)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_9)))) # label(fact_3217_of__nat__less__two__power) # label(axiom) # label(non_clause). [assumption]. 5.65/5.69 3483 (all A_44 all B_37 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_44),B_37)) -> hAPP_int_int(div_mod_int(B_37),A_44) = zero_zero_int)) # label(fact_2429_dvd__imp__mod__0) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3484 (all W_5 ((hAPP_int_nat(number_number_of_nat,W_5) = zero_zero_nat -> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,zero_zero_rat),hAPP_int_nat(number_number_of_nat,W_5)) = one_one_rat) & (hAPP_int_nat(number_number_of_nat,W_5) != zero_zero_nat -> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,zero_zero_rat),hAPP_int_nat(number_number_of_nat,W_5)) = zero_zero_rat))) # label(fact_1485_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3485 (all X_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> X_1 = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),N_1)))) # label(fact_3933_real__root__pow__pos2) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3486 (all N_1 N_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),N_1)) # label(fact_1793_nat__mult__1) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3487 (all X_1 all Y_1 X_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,fTrue),X_1),Y_1)) # label(help_If_1_1_If_000tc__Nat__Onat_T) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3488 (all N_1 all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(suc,M))) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),N_1)),hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1))) = hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),N_1)))) # label(fact_3777_fact__diff__Suc) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3489 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1)))))) # label(fact_3131_one__less__mult) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3490 (all A_106 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_106),zero_zero_complex) = A_106) # label(fact_1974_add__0__right) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3491 (all B_1_1 all B_2_1 is_int(hAPP_f659380387ol_int(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__Int__Oint) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3492 (all W all V_2 ((-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> hAPP_int_nat(nat,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,V_2)),hAPP_int_nat(number_number_of_nat,W))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,W))) & (hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),hAPP_int_nat(number_number_of_nat,W)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,W))))) # label(fact_3286_power__nat__number__of__number__of) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3493 (all A all B all C (hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A),B) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A),C) <-> C = B)) # label(fact_1938_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3494 (all I_1 (hAPP_int_nat(nat,I_1) = zero_zero_nat <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),zero_zero_int)))) # label(fact_2739_nat__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3495 (all D all A all B (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D),A) & hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D),B) = one_one_int <-> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),B)) = one_one_int)) # label(fact_5002_coprime__mul__eq__int) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3496 (all A_62 all B_54 all C_25 hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_62),B_54)),C_25) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_62),hAPP_nat_nat(div_mod_nat(B_54),C_25))),C_25)) # label(fact_2351_mod__add__right__eq) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3497 (all X_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,X_2)),one_one_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),hAPP_int_int(bit1,pls))))) # label(fact_290_le__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3498 (all X_2 hBOOL(hAPP_real_bool(isCont_real_real(cos),X_2))) # label(fact_5160_isCont__cos) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3499 (all X_1 all Y_1 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) -> X_1 != Y_1)) # label(fact_1707_dvd_Oless__imp__neq) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3500 (all Z_1 hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_complex_real(im,Z_1)),hAPP_complex_real(norm_norm_complex,Z_1)) = hAPP_complex_real(im,hAPP_complex_complex(sgn_sgn_complex,Z_1))) # label(fact_4104_Im__sgn) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3501 (all Z_8 all X_25 all Y_19 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,X_25),Y_19)) -> (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,X_25),Z_8)) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,X_25),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,Y_19),Z_8)))))) # label(fact_1529_dvd__diff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3502 (all A all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),zero_zero_int),P_5)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_5),A)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(big_co1548731110nt_int(cOMBI_int,comple219730294t_bool(setS(A,P_5)))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),P_5))))))) # label(fact_4808_Union__SetS__setprod__prop1) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3503 (all A all E_2 all C all B all D (is_int(D) -> (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),E_2)),D) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),E_2)),C) <-> D = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B)),E_2)),C)))) # label(fact_1551_eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3504 (all P_5 hAPP_P1175774780nt_int(produc1298267108nt_int(div_div_int),quotient_of(P_5)) = archim791455193or_rat(P_5)) # label(fact_4832_rat__floor__code) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3505 (all K_2 all M all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_2),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)) # label(fact_1645_Nat_Odiff__cancel) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3506 (all C_37 all A_94 all B_77 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_94),B_77)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_94),C_37)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_77),C_37))))) # label(fact_2060_add__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3507 (all D all F all X_2 all L (hBOOL(hAPP_real_bool(deriv_real(F,X_2),L)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D)) -> ((all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_2),Y))),D)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(F,Y)),hAPP_real_real(F,X_2))))) -> L = zero_zero_real)))) # label(fact_3883_DERIV__local__max) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3508 (all Y_1 all U_1 all X1_1 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X1_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X1_1),Y_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),U_1))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(inverse_inverse_real,X_1)),Y_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(inverse_inverse_real,X1_1)),U_1))))))) # label(fact_3836_real__mult__inverse__cancel) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3509 (all K_1 all Ma all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),Ma)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),N)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Ma),N))))) # label(fact_4327_gcd__greatest__iff__nat) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3510 (all A all B all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),hAPP_int_int(uminus_uminus_int,Ma))))) # label(fact_3518_zcong__zminus) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3511 (all B_1_1 is_int(archim856651990g_real(B_1_1))) # label(gsy_c_Archimedean__Field_Oceiling_000tc__RealDef__Oreal) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3512 (all C all A all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),C)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B),A)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,C),A))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B),C)))))) # label(fact_2552_div__dvd__div) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3513 (all N_1 N_1 = natceiling(hAPP_nat_real(real_nat,N_1))) # label(fact_3638_natceiling__real__of__nat) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3514 (all Lx all Rx all Ry hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx),Ry)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx),Ry))) # label(fact_179_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3515 (all X_1 all Y_1 (hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = Y_1 -> zero_zero_real = X_1)) # label(fact_3347_real__sqrt__sum__squares__eq__cancel2) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3516 (all X_1 all N_1 all Y_1 (hAPP_nat_nat(div_mod_nat(Y_1),N_1) = hAPP_nat_nat(div_mod_nat(X_1),N_1) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_1),X_1)) -> (exists Q_4 X_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),Q_4)))))) # label(fact_2529_nat__mod__eq__lemma) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3517 (all X_1 all Y_1 hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_1),Y_1) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_1),hAPP_complex_complex(uminus473333897omplex,Y_1))) # label(fact_3527_complex__diff__def) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3518 (all X_1 all K_2 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> hAPP_real_real(hAPP_n546711566l_real(root,N_1),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),K_2)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),K_2))) # label(fact_3908_real__root__power) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3519 (all Z_1 all W (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),Z_1)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_1),W)))) # label(fact_1522_real__le__linear) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3520 (all P_1 all A all B hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A),B) = hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),zero_zero_nat)) # label(fact_4243_Bolzano__bisect_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3521 (all P all Q all R hAPP_P877703507t_bool(hAPP_f1965248563t_bool(hAPP_f1273270095t_bool(cOMBB_595786433nt_int,P),Q),R) = hAPP_f513299955t_bool(P,hAPP_P921862901l_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J_000tc__fun_Itc_040) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3522 (all Y_1 exists X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X)) & Y_1 = hAPP_real_real(tan,X) & (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) & hAPP_real_real(tan,Y) = Y_1 & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),Y)) -> X = Y)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_3626_tan__total) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3523 (all K_2 K_2 = hAPP_i1732201573de_int(quickcheck_of_int,quickcheck_int_of(K_2))) # label(fact_4555_of__int__int__of) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3524 (all C_65 all D_12 all A_131 all B_110 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_131),B_110)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_65),D_12)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_131)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_65)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_131),C_65)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_110),D_12)))))))) # label(fact_1415_mult__strict__mono_H) # label(axiom) # label(non_clause). [assumption]. 5.69/5.69 3525 (all A_133 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_133),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_133),one_one_nat)))) # label(fact_1369_less__add__one) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3526 (all A_3 all B_2 hAPP_rat_rat(abs_abs_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(abs_abs_int,A_3)),hAPP_int_int(abs_abs_int,B_2))) # label(fact_4690_abs__rat) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3527 (all K_2 (is_int(K_2) -> K_2 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),pls))) # label(fact_357_add__Pls__right) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3528 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> M = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1)),N_1))) # label(fact_2734_div__mult__self__is__m) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3529 (all A_209 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_209),zero_z891286103de_int) = zero_z891286103de_int) # label(fact_620_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3530 (all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N))))) # label(fact_3150_of__nat__0__less__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3531 (all X_1 hAPP_int_nat(nat,archim1246769320r_real(X_1)) = natfloor(X_1)) # label(fact_4020_natfloor__def) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3532 (all X_2 all N all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),Y_2))) <-> hAPP_int_int(div_mod_int(X_2),N) = hAPP_int_int(div_mod_int(Y_2),N))) # label(fact_2440_zmod__eq__dvd__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3533 (all A all Ma all X_2 (is_int(X_2) -> (zero_zero_int != X_2 -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),Ma)) -> big_co1548731110nt_int(cOMBI_int,image_int_int(hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,times_times_int),X_2),hAPP_i1948725293t_bool(bnorRset(A),Ma))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,big_co1548731110nt_int(cOMBI_int,hAPP_i1948725293t_bool(bnorRset(A),Ma))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_2),hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(bnorRset(A),Ma)))))))) # label(fact_4811_Bnor__prod__power) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3534 (all X_1 all A_3 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,archim856651990g_real(X_1)),A_3) = archim856651990g_real(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),hAPP_int_real(real_int,A_3)))) # label(fact_4428_ceiling__subtract) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3535 (all X_29 all Y_23 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_29),Y_23)) -> (Y_23 != X_29 -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_29),Y_23))))) # label(fact_1014_order__le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3536 (all A_73 zero_z126310315umeral = hAPP_C498520661umeral(div_mo1740067990umeral(zero_z126310315umeral),A_73)) # label(fact_2312_mod__0) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3537 (all M all N_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(div_mod_nat(M),N_1)),M))) # label(fact_2508_mod__less__eq__dividend) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3538 (all B_71 all A_82 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_82)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_71)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_82),B_71)))))) # label(fact_2177_add__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3539 (all M all N_1 all Z_1 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,N_1)),Z_1)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1))),Z_1)) # label(fact_3043_zadd__int__left) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3540 (all X_1 all Y_1 all R_3 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,X_1),Y_1)),of_real_complex(R_3)) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),R_3)),Y_1)) # label(fact_3992_Complex__add__complex__of__real) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3541 (all A_28 (A_28 != zero_zero_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_28),A_28) = one_one_nat)) # label(fact_2586_div__self) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3542 (all B_2 all D_3 all A_3 (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),A_3) -> (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),B_2) -> one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2))))) # label(fact_5000_coprime__mult__int) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3543 (all A_31 all B_27 (zero_zero_nat != B_27 -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_27),A_31)),B_27) = A_31)) # label(fact_2574_div__mult__self1__is__id) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3544 (all A_107 A_107 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_107),zero_zero_nat)) # label(fact_1965_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3545 (all A all B all C (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C),zero_zero_rat)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),A)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),B))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B),A))))) # label(fact_1406_mult__le__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3546 (all M all V_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),V_2)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(pred,V_2))))) # label(fact_4050_Suc__diff__number__of) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3547 (all X_1 hAPP_int_nat(nat,archim856651990g_real(X_1)) = natceiling(X_1)) # label(fact_4363_natceiling__def) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3548 (all X_2 hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),cos_coeff)),hAPP_r474017924t_real(power_power_real,X_2))))) # label(fact_4831_summable__cos) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3549 (all V_8 all W_4 hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,V_8),W_4)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_int_complex(number528085621omplex,V_8)),hAPP_int_complex(number528085621omplex,W_4))) # label(fact_1518_number__of__diff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3550 (all A_222 (is_int(A_222) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_222),hAPP_int_int(number_number_of_int,pls)) = A_222)) # label(fact_417_add__numeral__0__right) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3551 (all X_3 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma)) -> (hBOOL(hAPP_f448129468l_bool(resSet(Ma),X_3)) -> hBOOL(hAPP_f448129468l_bool(finite_finite_int,X_3))))) # label(fact_4052_ResSet__finite) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3552 (all P_1 all N all K_1 (is_int(N) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N),K_1)),hAPP_int_int(div_mod_int(N),K_1))) <-> (all I_2 all J_1 (is_int(J_1) & is_int(I_2) -> (N = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),I_2)),J_1) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J_1),zero_zero_int)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),J_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,I_2),J_1))))))))) # label(fact_2700_split__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3553 (all B_1_1 is_bool(cauchy_complex(B_1_1))) # label(gsy_c_SEQ_OCauchy_000tc__Complex__Ocomplex) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3554 (all A_119 all B_98 all V_6 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_119),B_98)),hAPP_int_rat(number_number_of_rat,V_6)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_119),hAPP_int_rat(number_number_of_rat,V_6))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_98),hAPP_int_rat(number_number_of_rat,V_6)))) # label(fact_1562_left__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3555 (all B_126 all A_150 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_150)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_126),zero_zero_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_150),B_126)),zero_zero_rat))))) # label(fact_1238_mult__nonneg__nonpos) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3556 (all Z_1 Z_1 = hAPP_complex_real(re,of_real_complex(Z_1))) # label(fact_4113_Re__complex__of__real) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3557 (all N_1 all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)),N_1) = M)) # label(fact_1673_le__add__diff__inverse2) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3558 (all A_31 all B_27 (B_27 != zero_z126310315umeral -> hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_27),A_31)),B_27) = A_31)) # label(fact_2575_div__mult__self1__is__id) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3559 (exists X (hAPP_real_real(cos,X) = zero_zero_real & (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y)) & hAPP_real_real(cos,Y) = zero_zero_real & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) -> Y = X)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)))) # label(fact_3592_cos__is__zero) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3560 (all M_10 all N_24 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_10),N_24)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(semiri1424489471de_int,M_10)),hAPP_n522471361de_int(semiri1424489471de_int,N_24))))) # label(fact_2939_less__imp__of__nat__less) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3561 (all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> one_one_real = hAPP_real_real(hAPP_n546711566l_real(root,N_1),one_one_real))) # label(fact_3910_real__root__one) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3562 (all A_105 A_105 = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,zero_zero_real),A_105)) # label(fact_1983_add__0) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3563 (all N_1 one_one_real = hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_nat_real(real_nat,N_1))),pi))) # label(fact_3734_cos__2npi) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3564 (all N_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(suc,N_1)))) # label(fact_2821_zero__less__Suc) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3565 (all X_2 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Y_2)) -> (exists X (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X),field_1210416355s_real)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y_2)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),X)))))) # label(fact_4636_Rats__dense__in__real) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3566 (all X_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),X_1))) # label(fact_1636_dvd_Oorder__refl) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3567 (all A hBOOL(hAPP_complex_bool(isCont_complex_real(re),A))) # label(fact_5166_Re_OisCont) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3568 (all W_16 hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit1,pls)),W_16)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(number_number_of_int,W_16))) # label(fact_50_add__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3569 (all X_2 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),zero_zero_real)) -> hBOOL(hAPP_real_bool(isCont_real_real(hAPP_n546711566l_real(root,N)),X_2))))) # label(fact_5170_isCont__root__neg) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3570 (all Y_9 all Z_6 all X_13 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,Z_6),X_13)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,Z_6),Y_9)) -> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,X_13),Z_6)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,Y_9),Z_6)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X_13),Y_9)),Z_6)))) # label(fact_2610_div__add) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3571 (all B_2 all Q_8 all R_5 all Q_2 all R_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_8)),R_5)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_2)),R_3))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_5),B_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_8),Q_2))))))) # label(fact_1497_unique__quotient__lemma) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3572 (all P all Q all R hAPP_r337325687l_real(hAPP_f1732848529l_real(hAPP_f1310479593l_real(cOMBB_2092576745l_real,P),Q),R) = hAPP_f556082547l_real(P,hAPP_r1250527377l_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc_) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3573 (all X_1 all Y_1 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,ratreal(X_1)),ratreal(Y_1)) = ratreal(hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_1),Y_1))) # label(fact_4704_real__times__code) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3574 (all X_2 all A (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(real_int,A)),one_one_real))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim856651990g_real(X_2)),A)))) # label(fact_4470_ceiling__less__eq) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3575 (all A_12 all B_9 A_12 = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_12),B_9)),B_9)),hAPP_C498520661umeral(div_mo1740067990umeral(A_12),B_9))) # label(fact_2710_mod__div__decomp) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3576 (all N all Ma (Ma = N <-> hAPP_nat_real(real_nat,N) = hAPP_nat_real(real_nat,Ma))) # label(fact_3635_real__of__nat__inject) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3577 (all X_49 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_49),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_49),X_49)) # label(fact_34_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3578 (all K_2 all L_2 (zero_zero_int != hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,K_2)),L_2) -> hAPP_int_int(div_mod_int(K_2),L_2) != zero_zero_int)) # label(fact_3532_zmod__zminus1__not__zero) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3579 (all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,Y_1)),one_one_real)) -> hAPP_real_real(sin,hAPP_real_real(arccos,Y_1)) = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_3742_sin__arccos__abs) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3580 (all V_8 all W_4 hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,V_8),W_4)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_int_rat(number_number_of_rat,V_8)),hAPP_int_rat(number_number_of_rat,W_4))) # label(fact_1519_number__of__diff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3581 (all P all Q all R hAPP_i1584592887nt_int(hAPP_f1574055885nt_int(hAPP_f434654626nt_int(cOMBS_2011279539nt_int,P),Q),R) = hAPP_i1584592887nt_int(hAPP_i1778081398nt_int(P,R),hAPP_int_int(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__Int__Oint_000tc__fun_Itc__Int__Oint) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3582 (all B_112 all C_67 all A_134 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_134)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_112),C_67)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_112),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_134),C_67)))))) # label(fact_1335_pos__add__strict) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3583 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (all Y exists N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),X_1)))))) # label(fact_3750_reals__Archimedean3) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3584 (all A_194 all B_153 all N_46 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_194),N_46)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_153),N_46)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_194),B_153)),N_46)) # label(fact_816_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3585 (all P all Q all R hAPP_f1735705551nt_int(hAPP_i1836591008nt_int(P,R),hAPP_i1584592887nt_int(Q,R)) = hAPP_i1584592887nt_int(hAPP_f465821005nt_int(hAPP_f1081447804nt_int(cOMBS_1470252563nt_int,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__prod_Itc__I) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3586 (all K_2 all L_2 hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,hAPP_int_rat(ring_1_of_int_rat,K_2)),hAPP_int_rat(ring_1_of_int_rat,L_2)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,K_2),L_2)) # label(fact_4679_Fract__of__int__quotient) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3587 (all X_23 hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_23),X_23)),one_one_rat) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_23),one_one_rat)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_23),one_one_rat))) # label(fact_1546_real__squared__diff__one__factored) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3588 (all B_36 zero_zero_complex = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,zero_zero_complex),B_36)) # label(fact_2471_mult_Ozero__left) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3589 (all A_45 all C_13 all B_38 hAPP_nat_nat(div_mod_nat(A_45),B_38) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_45),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_13),B_38))),B_38)) # label(fact_2424_mod__mult__self1) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3590 (all D all Sa exists Z all X (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Z),X)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,D),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X),Sa))) <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,D),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X),Sa)))))) # label(fact_2247_pinf_I9_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3591 (all A (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),A)),zero_zero_real)))) # label(fact_688_even__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3592 (all X_2 all Y_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_2)) | Y_2 = X_2 <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2)))) # label(fact_1692_dvd_Ole__less) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3593 (all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(sqrt,Y_2))))) # label(fact_3334_real__sqrt__gt__1__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3594 (all P all Q all R hAPP_i1778081398nt_int(hAPP_f2094497995nt_int(hAPP_f534361363nt_int(cOMBB_1657320178nt_int,P),Q),R) = hAPP_f465821005nt_int(P,hAPP_i1778081398nt_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__prod) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3595 (all P all Q all R hAPP_i1823918693l_bool(hAPP_f1344929775l_bool(hAPP_f1830214065l_bool(cOMBB_793166024ol_int,P),Q),R) = hAPP_f2144054103l_bool(P,hAPP_i1948725293t_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__020) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3596 (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,X_2)),one_one_real)) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_f1122023986l_real(hAPP_f1786887945l_real(cOMBB_1896127834l_real,suminf_real),hAPP_f1993765077t_real(hAPP_f895854793t_real(cOMBB_1948233853l_real,hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))))),hAPP_f1993765077t_real(hAPP_f895854793t_real(cOMBB_1948233853l_real,hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,one_one_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat)))))),hAPP_f1501741374t_real(hAPP_f1135648319t_real(cOMBC_1329014627t_real,hAPP_f1163894827t_real(hAPP_f1998755145t_real(cOMBB_421464275l_real,cOMBB_nat_real_nat),power_power_real)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))),X_2),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_2)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))))) # label(fact_4817_DERIV__arctan__series) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3597 (all C_33 all D_5 all A_78 all B_67 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_78),B_67)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,C_33),D_5)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_78),C_33)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_67),D_5)))))) # label(fact_2208_add__less__le__mono) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3598 (all A all N (is_int(A) -> (zero_zero_nat != N & A = zero_zero_int <-> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),N) = zero_zero_int))) # label(fact_546_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3599 (all K_2 all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_2),K_2)))))) # label(fact_1800_mult__less__mono1) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3600 (all C_88 all A_180 all B_152 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_180),B_152)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_152),C_88)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_180),C_88))))) # label(fact_990_dvd__trans) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3601 (all M all N_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N_1)),N_1))) # label(fact_4297_gcd__dvd2__nat) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3602 (all M all N_1 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_nat_real(real_nat,M)),N_1) = hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M),N_1))) # label(fact_3654_real__of__nat__power) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3603 (all P_1 all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) -> (all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_2))),hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_2))))))) # label(fact_4242_Bolzano__bisect__le) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3604 (all V_10 all U_3 all Y_25 all X_33 all A_185 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_33),A_185)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_25),A_185)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),U_3)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),V_10)) -> (one_one_real = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,U_3),V_10) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,U_3),X_33)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,V_10),Y_25))),A_185)))))))) # label(fact_907_convex__bound__lt) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3605 (all Ma all N (N = zero_zero_nat & Ma = zero_zero_nat <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),N) = zero_zero_nat)) # label(fact_1746_add__is__0) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3606 (all B_1_1 all B_2_1 (is_bool(B_2_1) & is_bool(B_1_1) -> is_bool(fdisj(B_1_1,B_2_1)))) # label(gsy_c_fdisj) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3607 (all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),one_one_real)) -> (exists X (Y_1 = hAPP_real_real(cos,X) & (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),pi)) & hAPP_real_real(cos,Y) = Y_1 & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y)) -> Y = X)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),pi)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))))))) # label(fact_3591_cos__total) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3608 (all M_11 -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,M_11)),zero_zero_real))) # label(fact_2921_of__nat__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3609 (all P all Q all R hAPP_f1908301369l_real(P,hAPP_r1230691443l_real(Q,R)) = hAPP_r224952523l_real(hAPP_f1663701695l_real(hAPP_f774948957l_real(cOMBB_3698607l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__fun_Itc__prod_Itc__Real) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3610 (all V_8 all W_4 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(number267125858f_real,V_8)),hAPP_int_real(number267125858f_real,W_4)) = hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,V_8),W_4))) # label(fact_1517_number__of__diff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3611 (all Y_1 all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_1),X_1)) -> hAPP_int_int(nat_tsub(X_1),Y_1) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),Y_1))) # label(fact_2798_tsub__eq) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3612 (all N_1 all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N_1))) # label(fact_4358_gcd__diff1__nat) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3613 (all A_215 all M_24 all N_64 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_24),N_64)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_215),M_24)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_215),N_64))))) # label(fact_555_le__imp__power__dvd) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3614 (all N_1 all M (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),M)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(suc,M))) -> N_1 = M))) # label(fact_2861_less__antisym) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3615 (all A_3 all B_2 (is_int(B_2) -> (zero_zero_int != B_2 -> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),divmod_int(A_3,B_2)))))) # label(fact_4191_divmod__int__correct) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3616 (all A_187 all N_37 all N_36 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_37),N_36)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_187)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_187),N_37)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_187),N_36)))))) # label(fact_890_power__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3617 (all K_2 all L_2 (hAPP_int_int(div_mod_int(K_2),hAPP_int_int(uminus_uminus_int,L_2)) != zero_zero_int -> zero_zero_int != hAPP_int_int(div_mod_int(K_2),L_2))) # label(fact_3533_zmod__zminus2__not__zero) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3618 (all R_3 all X_1 hAPP_real_real(scaleR_scaleR_real(R_3),hAPP_complex_real(im,X_1)) = hAPP_complex_real(im,hAPP_complex_complex(scaleR1652505878omplex(R_3),X_1))) # label(fact_5111_Im_OscaleR) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3619 (all V all V_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_int_nat(number_number_of_nat,V_1))) <-> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V),V_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V),pls))))) # label(fact_741_le__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3620 (all U_1 all M all N_1 all J_2 all I (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),I)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),U_1)),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_2),U_1)),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),J_2)),U_1)),M)),N_1))) # label(fact_1851_nat__diff__add__eq1) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3621 (all P all Q all R hAPP_r50993370t_real(hAPP_f1163894827t_real(hAPP_f1998755145t_real(cOMBB_421464275l_real,P),Q),R) = hAPP_f1566856189t_real(P,hAPP_r474017924t_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_000tc__fun__029) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3622 (all A hBOOL(hAPP_real_bool(isCont_real_real(abs_abs_real),A))) # label(fact_5167_isCont__abs) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3623 (all C_75 all A_156 all B_132 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_156),B_132)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_75)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_75),A_156)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_75),B_132)))))) # label(fact_1202_comm__mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3624 (all C_59 all A_125 all B_104 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_59),A_125)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_59),B_104))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_59)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_125),B_104))))) # label(fact_1453_mult__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3625 (all M all N_1 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1) = zero_zero_nat -> (zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),M) -> N_1 = M))) # label(fact_1637_diffs0__imp__equal) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3626 (all X_1 all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),X_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),one_one_int),P_3))))))) # label(fact_2714_Euler__part3) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3627 (all A_3 all B_2 (is_int(B_2) -> (zero_zero_int != B_2 -> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),hAPP_int_int(div_mod_int(A_3),B_2))))))) # label(fact_2799_divmod__int__rel__div__mod) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3628 (all X_1 all N_1 all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),N_1))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),N_1)))))) # label(fact_4172_prime__divexp__n) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3629 (all A_10 all N_17 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_10),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_10),N_17)) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_10),hAPP_nat_nat(suc,N_17))) # label(fact_3035_power__Suc) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3630 (all A_91 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_91),one_one_rat) = A_91) # label(fact_2106_mult_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3631 (all N_66 all A_217 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_217)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_217),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_217),N_66)))))) # label(fact_524_power__gt1__lemma) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3632 (all X_1 all Y_1 Y_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,fFalse),X_1),Y_1)) # label(help_If_2_1_If_000tc__Nat__Onat_T) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3633 (all B_72 all C_35 all A_83 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_83)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_72),C_35)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_72),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_83),C_35)))))) # label(fact_2163_add__increasing) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3634 (all I all K_2 all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_2)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J_2)),K_2))) # label(fact_1675_add__diff__assoc) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3635 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,X_1)),X_1)))) # label(fact_3559_ln__bound) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3636 (all Ta all B_1 all D_1 (is_int(Ta) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Ta),one_one_int)),B_1)) -> (all X (is_int(X) -> ((all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B_1)) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) != X)))))) -> (X = Ta -> Ta = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),D_1))))))))) # label(fact_5092_bset_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3637 (all A_139 all B_116 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_139),B_116))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_139)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_116))))) # label(fact_1299_zero__less__mult__pos) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3638 (all X_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_2)) <-> hBOOL(hAPP_int_bool(even_odd_even_int,X_2)))) # label(fact_3223_int__even__iff__2__dvd) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3639 (all N_27 all Y_8 all X_12 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,Y_8),X_12)) -> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_12),N_27)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_8),N_27)) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,X_12),Y_8)),N_27))) # label(fact_2618_div__power) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3640 (all D all Sa exists Z ((all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),Z)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Sa))) <-> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Sa)))))) & is_int(Z))) # label(fact_2267_minf_I10_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3641 (all A_35 hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,zero_zero_nat),A_35) = zero_zero_nat) # label(fact_2542_div__0) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3642 (all B_138 all A_162 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_162),zero_zero_real)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_138)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_138),zero_zero_real)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_162)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_162),B_138)),zero_zero_real)))) # label(fact_1175_split__mult__neg__le) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3643 (all X_2 all Y_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2)) & Y_2 != X_2 <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_2)))) # label(fact_1693_dvd_Oless__le) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3644 (all X_47 all P_8 all Q_12 hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_47),P_8)),Q_12) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_47),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_8),Q_12))) # label(fact_70_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3645 (all P all Q all R hAPP_f1293550189l_real(hAPP_r1043989325l_real(P,R),hAPP_r2019696015l_real(Q,R)) = hAPP_r1230691443l_real(hAPP_f324002831l_real(hAPP_f677014621l_real(cOMBS_107495110l_real,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__p_042) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3646 (all A_238 all B_167 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_167),A_238) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_238),B_167)) # label(fact_190_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3647 (all A_233 all C_93 all D_18 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_233),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_93),D_18)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_93),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_233),D_18))) # label(fact_236_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3648 (all N all X_2 ((hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2)) -> hAPP_real_real(hAPP_n546711566l_real(root,N),X_2) = the_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_real,zero_zero_real))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,power_power_real),N))),X_2)))) & (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2)) -> (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),zero_zero_real)) -> hAPP_real_real(hAPP_n546711566l_real(root,N),X_2) = zero_zero_real) & (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),zero_zero_real)) -> hAPP_real_real(hAPP_n546711566l_real(root,N),X_2) = hAPP_real_real(uminus_uminus_real,the_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_real,zero_zero_real))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,power_power_real),N))),hAPP_real_real(uminus_uminus_real,X_2))))))))) # label(fact_4884_root__def) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3649 (all C_63 all D_10 all A_129 all B_108 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_129),B_108)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_63),D_10)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_129)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_63)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_129),C_63)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_108),D_10)))))))) # label(fact_1425_mult__le__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3650 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K1)),hAPP_int_int(bit1,K2))))) # label(fact_108_less__int__code_I16_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3651 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(ln,X_1))))) # label(fact_3438_ln__ge__zero) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3652 (all C_75 all A_156 all B_132 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_156),B_132)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_75)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_75),A_156)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_75),B_132)))))) # label(fact_1201_comm__mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3653 (all Z_8 all X_25 all Y_19 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_25),Y_19)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_25),Z_8)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_25),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_19),Z_8)))))) # label(fact_1528_dvd__diff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3654 (all B_115 all A_138 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_138)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_115),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_115),A_138)),zero_zero_real))))) # label(fact_1304_mult__pos__neg2) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3655 (all P all Q all R hAPP_b589554111l_bool(P,hAPP_nat_bool(Q,R)) = hAPP_n1006566506l_bool(hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_005) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3656 (all N_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_1)) -> hAPP_int_int(fact_fact_int,N_1) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N_1),hAPP_int_int(fact_fact_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,N_1),one_one_int))))) # label(fact_4287_fact__reduce__int) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3657 (all A_20 all B_17 (B_17 != zero_z891286103de_int -> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_20),B_17)),B_17) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_20),B_17)),one_on1684967323de_int))) # label(fact_2654_div__add__self2) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3658 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> (hAPP_real_real(cos,X_1) = zero_zero_real -> (exists N_2 (X_1 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) & -hBOOL(hAPP_nat_bool(even_odd_even_nat,N_2))))))) # label(fact_3793_cos__zero__lemma) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3659 (all X_1 all Y_1 hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_1),Y_1)) = hAPP_complex_real(dist_dist_complex(X_1),Y_1)) # label(fact_4545_dist__complex__def) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3660 (all A_3 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_2),one_one_int))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),B_2)))) # label(fact_1838_norR__mem__unique__aux) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3661 (all Ma all N (N = Ma <-> hAPP_nat_int(semiri1621563631at_int,N) = hAPP_nat_int(semiri1621563631at_int,Ma))) # label(fact_2833_of__nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3662 (all A_3 (is_int(A_3) -> (A_3 != zero_zero_int -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),A_3) = one_one_int))) # label(fact_2558_zdiv__self) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3663 (all A_203 all N_53 all B_155 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_203),N_53)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_155),N_53))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_155)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_203),B_155))))) # label(fact_671_power__less__imp__less__base) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3664 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),zero_zero_real)) -> hAPP_real_real(sgn_sgn_real,X_1) = hAPP_int_real(number267125858f_real,min))) # label(fact_3573_real__sgn__neg) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3665 (all X_48 all N_69 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_48),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_69)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_48),N_69)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_48),N_69))) # label(fact_47_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3666 (all A_208 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,zero_zero_complex),A_208) = A_208) # label(fact_626_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3667 (all A_194 all B_153 all N_46 hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_194),B_153)),N_46) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_194),N_46)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_153),N_46))) # label(fact_811_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3668 (all X_2 all Y_2 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_2)),Y_2)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_2),Y_2)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_12_power2__sum) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3669 (all A_196 -hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_196),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),zero_zero_rat))) # label(fact_797_power2__less__0) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3670 (all Lx_4 all Ly_2 all Rx_4 all Ry_2 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_4),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Ly_2),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Rx_4),Ry_2))) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_4),Ly_2)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Rx_4),Ry_2))) # label(fact_150_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3671 (all R_3 rcis(R_3,zero_zero_real) = of_real_complex(R_3)) # label(fact_4257_rcis__zero__arg) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3672 (all B_55 all A_63 hAPP_C498520661umeral(div_mo1740067990umeral(A_63),B_55) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_55),A_63)),B_55)) # label(fact_2348_mod__add__self1) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3673 (all A all B ((all D_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),A)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),B)) <-> D_2 = one_one_nat)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A),B)))) # label(fact_4027_coprime) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3674 (all P all Q (hBOOL(Q) | -hBOOL(P) | -hBOOL(hAPP_bool_bool(fimplies(P),Q)))) # label(help_fimplies_3_1_U) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3675 (all A_1 (hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1)) -> hAPP_f22106695ol_nat(finite_card_nat,image_int_nat(nat,A_1)) = hAPP_f957591787ol_nat(finite_card_int,A_1))) # label(fact_4619_Nat__Transfer_Otransfer__int__nat__set__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3676 (all V_11 all W_9 hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_11),W_9)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_11)),hAPP_int_real(number267125858f_real,W_9))) # label(fact_442_number__of__add) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3677 (all A_229 all B_162 all C_91 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_229),B_162)),C_91) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_229),C_91)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_162),C_91))) # label(fact_311_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3678 (all P all Q all R hAPP_int_int(P,hAPP_P1175774780nt_int(Q,R)) = hAPP_P1175774780nt_int(hAPP_f521271395nt_int(hAPP_f11151005nt_int(cOMBB_47643171nt_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__Int__Oint_000tc__prod_Itc__Int__Oin) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3679 (all X_1 (hBOOL(hAPP_int_bool(even_odd_even_int,X_1)) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3244_even__plus__one__div__two) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3680 (all B all A all C (is_int(B) & is_int(C) -> (B = C <-> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C),A) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B),A)))) # label(fact_1932_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3681 (all N_1 hAPP_complex_real(re,hAPP_nat_complex(semiri2020571505omplex,N_1)) = hAPP_nat_real(semiri132038758t_real,N_1)) # label(fact_4126_complex__Re__of__nat) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3682 (all S ((all X (is_int(X) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X),S)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))))) <-> hBOOL(hAPP_f448129468l_bool(nat_nat_set,S)))) # label(fact_4612_nat__set__def) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3683 (all X_1 all Y_1 X_1 = hAPP_real_real(hAPP_r1250527377l_real(hAPP_b646336293l_real(if_real,fTrue),X_1),Y_1)) # label(help_If_1_1_If_000tc__RealDef__Oreal_T) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3684 (all K_2 all L_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),L_2)) -> (exists N_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_2),N_2) = L_2))) # label(fact_3952_le__Suc__ex) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3685 (all B_1_1 all B_2_1 is_bool(hAPP_C24501650l_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__Code____Numeral__Ocode____numeral_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3686 (all X_44 all Y_36 all Z_15 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_44),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Y_36),Z_15)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_44),Y_36)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_44),Z_15))) # label(fact_321_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3687 (all A all E_2 all C all B all D (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B)),E_2)),C)),D)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),E_2)),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),E_2)),D))))) # label(fact_1579_less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3688 (all A_93 hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_93),zero_zero_complex) = A_93) # label(fact_2080_diff__0__right) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3689 (all X_1 all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),X_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_int_int(number_number_of_int,min)),P_3))))))) # label(fact_2713_Euler__part1) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3690 (all Ma all N all A (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A)) -> (Ma = N <-> hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A),Ma) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A),N)))) # label(fact_875_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3691 (all A all B all C (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),A)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),A)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B)))))) # label(fact_1287_mult__less__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3692 (all A_86 all B_75 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_86),B_75)),B_75) = A_86) # label(fact_2139_diff__add__cancel) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3693 (all X_2 (one_one_rat = X_2 <-> one_one_rat = X_2)) # label(fact_1958_one__reorient) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3694 (all P all Q all R hAPP_P989584703t_bool(hAPP_f1781059817t_bool(hAPP_f125837377t_bool(cOMBS_1610675060t_bool,P),Q),R) = hAPP_f237669397t_bool(hAPP_P877703507t_bool(P,R),hAPP_P989584703t_bool(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__033) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3695 (all A_50 all C_15 all B_42 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C_15),B_42)) -> hAPP_nat_nat(div_mod_nat(A_50),C_15) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(div_mod_nat(A_50),B_42)),C_15))) # label(fact_2389_mod__mod__cancel) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3696 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(uminus_uminus_real,X_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),X_1))))))) # label(fact_3503_ln__one__minus__pos__lower__bound) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3697 (all Ma all K_1 all N (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,K_1),N)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,K_1),Ma)) <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,K_1),hAPP_Q1762011733de_int(div_mo231679042de_int(Ma),N)))))) # label(fact_2391_dvd__mod__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3698 (all A_238 all B_167 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_238),B_167) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_167),A_238)) # label(fact_189_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3699 (all A_172 all B_147 all C_85 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_172),B_147)),C_85)) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,B_147),C_85)))) # label(fact_1087_dvd__mult__right) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3700 (all X_11 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_11)) -> hAPP_nat_nat(semiri984289939at_nat,hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_11))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(semiri984289939at_nat,hAPP_int_nat(nat,X_11))),hAPP_nat_nat(semiri984289939at_nat,hAPP_int_nat(nat,X_11))))) # label(fact_2811_of__nat__double) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3701 (all A_3 all B_2 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),B_2)),ii) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(uminus_uminus_real,B_2)),A_3)) # label(fact_3981_Complex__mult__i) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3702 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X_1))),X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))))) # label(fact_3476_abs__ln__one__plus__x__minus__x__bound__nonpos) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3703 (all A all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),zero_zero_int),P_5)) -> big_co1548731110nt_int(cOMBI_int,comple219730294t_bool(setS(A,P_5))) = hAPP_int_int(zfact,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)))))) # label(fact_4814_Union__SetS__setprod__prop2) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3704 (all A_202 all N_52 all N_51 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_52),N_51)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_202)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_202),one_on1645066479umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_202),N_51)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_202),N_52))))))) # label(fact_678_power__decreasing) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3705 (all X_2 all N (N = zero_zero_nat | X_2 = one_one_nat <-> one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_2),N))) # label(fact_1613_exp__eq__1) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3706 (all B all A (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(d22set,A))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A)))) # label(fact_2271_d22set__le) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3707 (all C_46 all A_103 all B_86 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_46),A_103)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_46),B_86))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_103),B_86)))) # label(fact_1996_add__le__imp__le__left) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3708 (all M_14 all D_4 (is_int(M_14) -> (zero_zero_int = hAPP_int_int(div_mod_int(M_14),D_4) -> (exists Q_4 (is_int(Q_4) & M_14 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,D_4),Q_4)))))) # label(fact_2504_zmod__eq__0D) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3709 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),M)),N_1) = M)) # label(fact_2735_div__mult__self1__is__m) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3710 (all A_139 all B_116 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_139),B_116))) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_139)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_116))))) # label(fact_1301_zero__less__mult__pos) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3711 (all P all Q all R hAPP_f627970963t_bool(P,hAPP_i68813070l_bool(Q,R)) = hAPP_i1529485324t_bool(hAPP_f1315737299t_bool(hAPP_f538565589t_bool(cOMBB_1211207212ol_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J_000tc__fun_Itc) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3712 (all B_125 all A_149 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_149)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_125)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_149),B_125)))))) # label(fact_1239_mult__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3713 (all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> -hBOOL(hAPP_nat_bool(even_odd_even_nat,P_3)) | P_3 = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_4195_prime__odd) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3714 (all B all C all A all D (hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A),D) = one_one_int -> (hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,B),C) = one_one_int -> (hAPP_int_int(abs_abs_int,B) = hAPP_int_int(abs_abs_int,A) & hAPP_int_int(abs_abs_int,C) = hAPP_int_int(abs_abs_int,D) <-> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,B)),hAPP_int_int(abs_abs_int,D)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,A)),hAPP_int_int(abs_abs_int,C)))))) # label(fact_5013_coprime__crossproduct__int) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3715 (all X_2 all Y_2 (hAPP_P1175774780nt_int(product_snd_int_int,Y_2) != zero_zero_int & hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_fst_int_int,X_2)),hAPP_P1175774780nt_int(product_snd_int_int,Y_2)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_fst_int_int,Y_2)),hAPP_P1175774780nt_int(product_snd_int_int,X_2)) & zero_zero_int != hAPP_P1175774780nt_int(product_snd_int_int,X_2) <-> hBOOL(hAPP_f526364711l_bool(hAPP_P1062890741l_bool(member2143287562nt_int,hAPP_P230595783nt_int(hAPP_P941956607nt_int(produc883642259nt_int,X_2),Y_2)),ratrel)))) # label(fact_4665_ratrel__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3716 (all I all J_2 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(legacy_zgcd(I),J_2)))) # label(fact_4496_zgcd__pos) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3717 (all L all U hAPP_f22106695ol_nat(finite_card_nat,hAPP_n1699378549t_bool(ord_gr660468384an_nat(L),U)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,U),hAPP_nat_nat(suc,L))) # label(fact_4632_card__greaterThanLessThan) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3718 (all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N))))) # label(fact_3152_zero__less__int__conv) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3719 (all A_23 all B_20 hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(div_mod_nat(A_23),B_20)),B_20) = zero_zero_nat) # label(fact_2614_mod__div__trivial) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3720 (all Y_2 all X_2 (is_int(X_2) & is_int(Y_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2)) -> (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Y_2) = zero_zero_int <-> X_2 = zero_zero_int & Y_2 = zero_zero_int))))) # label(fact_2167_add__nonneg__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3721 (all M all N_1 xzgcda(M,N_1,M,N_1,one_one_int,zero_zero_int,zero_zero_int,one_one_int) = xzgcd(M,N_1)) # label(fact_4200_xzgcd__def) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3722 (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(exp_real,X_2))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2)))) # label(fact_3841_one__less__exp__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3723 (all Sa all Ta all A all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A),B)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Sa),Ta)) -> hBOOL(hAPP_f115998247l_bool(hAPP_P1876494581l_bool(member180897546at_nat,hAPP_P369611207at_nat(hAPP_P402336767at_nat(produc494345619at_nat,hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,A),Sa)),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,B),Ta))),pair_leq))))) # label(fact_4541_pair__leqI2) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3724 (all X_34 all Y_26 -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_34),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_26),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),zero_zero_real))) # label(fact_852_not__sum__power2__lt__zero) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3725 (all K_1 all L (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,K_1)),hAPP_int_int(bit0,L))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L)))) # label(fact_128_rel__simps_I33_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3726 (all B_113 all A_136 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_136)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_113)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_136),B_113)))))) # label(fact_1315_mult__pos__pos) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3727 (all A_59 all B_51 all C_22 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_59),hAPP_int_int(div_mod_int(B_51),C_22))),C_22) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_59),B_51)),C_22)) # label(fact_2362_zmod__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3728 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sqrt,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = X_1)) # label(fact_3341_real__sqrt__pow2) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3729 (all N_67 all A_218 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_218)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_218),N_67)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_218),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_218),N_67)))))) # label(fact_518_power__less__power__Suc) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3730 (all X_1 all Y_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,X_1),Y_1)) | X_1 != Y_1)) # label(help_fequal_2_1_fequal_000tc__Int__Oint_T) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3731 (all A_91 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_91),one_one_complex) = A_91) # label(fact_2104_mult_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3732 (all X_1 all Y_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) | X_1 = Y_1)) # label(fact_1698_dvd_Ole__imp__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3733 (all X_21 all N_31 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31)) -> hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_21),N_31) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_21),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_31),one_one_nat))),X_21))) # label(fact_1733_realpow__minus__mult) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3734 (all N_59 all A_212 (zero_zero_real != A_212 -> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_212),N_59) != zero_zero_real)) # label(fact_591_field__power__not__zero) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3735 (all M all N_1 ((zero_zero_nat != hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1) -> hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1)),hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1)),one_one_nat)))) & (zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1) -> hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1)) = one_one_nat))) # label(fact_3799_fact__add__num__eq__if__nat) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3736 (all A_183 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_183),zero_zero_nat) = one_one_complex) # label(fact_959_power__0) # label(axiom) # label(non_clause). [assumption]. 5.69/5.70 3737 (all X_2 all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),hAPP_i1948725293t_bool(sRStar,P_5))) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(standardRes(P_5),X_2)),hAPP_i1948725293t_bool(sRStar,P_5))))))) # label(fact_2526_StandardRes__SRStar__prop4) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3738 (all X_1 all N_1 (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,N_1))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natfloor(X_1)),hAPP_int_nat(number_number_of_nat,N_1)) = natfloor(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_int_real(number267125858f_real,N_1)))))) # label(fact_3380_natfloor__add__number__of) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3739 (all A_139 all B_116 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_139),B_116))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_139)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_116))))) # label(fact_1296_zero__less__mult__pos) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3740 (all X_1 all Y_1 hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sqrt,X_1)),hAPP_real_real(sqrt,Y_1)) = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_1),Y_1))) # label(fact_3355_real__sqrt__divide) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3741 (all A_3 all Y_1 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1)) -> hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),Y_1)),A_3) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,Y_1),A_3))))) # label(fact_4430_powr__mult) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3742 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)),A_3)))) # label(fact_5009_gcd__le1__int) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3743 (all X_2 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2)) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(root,N),X_2),hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,zero_zero_nat)))))))))) # label(fact_3941_DERIV__real__root) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3744 (all X_1 all Y_1 all Z_1 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),Y_1)),Z_1) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Y_1),Z_1))) # label(fact_83_zpower__zpower) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3745 (all B all A all Ma (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(bnorRset(A),Ma))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)))) # label(fact_4294_Bnor__mem__zg) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3746 (all V all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_nat_nat(suc,N))) <-> (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))),N))))) # label(fact_4785_le__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3747 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X_1))))))) # label(fact_3472_ln__one__plus__pos__lower__bound) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3748 (all X_1 all Y_1 hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_1) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(abs_abs_int,X_1)),Y_1)) # label(fact_4991_gcd__abs1__int) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3749 (all N_1 archim1246769320r_real(hAPP_real_real(uminus_uminus_real,hAPP_int_real(real_int,N_1))) = hAPP_int_int(uminus_uminus_int,N_1)) # label(fact_4427_floor__minus__real__of__int) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3750 (all B_115 all A_138 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_138)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_115),zero_z891286103de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_115),A_138)),zero_z891286103de_int))))) # label(fact_1306_mult__pos__neg2) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3751 (all P all Q all R hAPP_i1524277240nt_int(hAPP_f1735705551nt_int(hAPP_f713949180nt_int(cOMBS_1891347093nt_int,P),Q),R) = hAPP_P1975530577nt_int(hAPP_i1216375074nt_int(P,R),hAPP_i1524277240nt_int(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3752 (all M one_one_int = hAPP_int_int(legacy_zgcd(M),one_one_int)) # label(fact_4485_zgcd__1) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3753 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N_1))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)))) # label(fact_2867_Suc__less__SucD) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3754 (all A_3 A_3 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),A_3)) # label(fact_4302_gcd__semilattice__nat_Oinf_Oidem) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3755 (all Z_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Z_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Z_1),Z_1)) # label(fact_106_nat__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3756 (all X_1 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_515_four__x__squared) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3757 (all N all F (hBOOL(summable_real(F)) -> ((all M_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(F,M_1))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_f352196356l_real(suminf_real,F)))))) # label(fact_4928_series__pos__less) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3758 (all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N)) -> hAPP_int_int(fact_fact_int,N) = big_co1548731110nt_int(cOMBI_int,hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),N)))) # label(fact_5083_fact__altdef__int) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3759 (all X_1 all Y_1 (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(fequal_real,X_1),Y_1)) | Y_1 = X_1)) # label(help_fequal_1_1_fequal_000tc__RealDef__Oreal_T) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3760 (all A_193 all M_21 all N_45 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_193),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_21),N_45)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_193),M_21)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_193),N_45))) # label(fact_824_power__add) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3761 (all X_2 all Y_2 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Y_2)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_2)),Y_2))) # label(fact_9_power2__sum) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3762 (all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),Y_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arcsin,Y_1))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arcsin,Y_1)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3505_arcsin__lt__bounded) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3763 (all A_3 all B_2 hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(uminus_uminus_int,A_3)),B_2) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(uminus_uminus_int,B_2))) # label(fact_3521_zdiv__zminus2) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3764 (all Y_2 all X_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) -> hBOOL(hAPP_f448129468l_bool(nat_nat_set,hAPP_i1948725293t_bool(ord_at875362053st_int(X_2),Y_2))))) # label(fact_5067_SetInterval_Otransfer__nat__int__set__function__closures) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3765 (all Z_1 all X_1 all Y_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Z_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y_1),Z_1)))))) # label(fact_4318_gcd__semilattice__nat_Oinf__greatest) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3766 (all A_220 A_220 = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_220),hAPP_int_complex(number528085621omplex,hAPP_int_int(bit1,pls)))) # label(fact_458_mult__numeral__1__right) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3767 (all X_47 all P_8 all Q_12 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_47),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_8),Q_12)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_47),P_8)),Q_12)) # label(fact_75_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3768 (all C_64 all D_11 all A_130 all B_109 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_130),B_109)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_64),D_11)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_130)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_64)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_130),C_64)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_109),D_11)))))))) # label(fact_1423_mult__less__le__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3769 (all B_1_1 is_int(hilbert_Eps_int(B_1_1))) # label(gsy_c_Hilbert__Choice_OEps_000tc__Int__Oint) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3770 (all K_2 hAPP_int_rat(hAPP_int_fun_int_rat(fract,zero_zero_int),K_2) = zero_zero_rat) # label(fact_4673_rat__number__collapse_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3771 (all C_1 all A_3 all B_2 all Q_2 all R_3 (hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(A_3,B_2),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Q_2),R_3))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_1)) -> hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(A_3,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),C_1)),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,Q_2),C_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),hAPP_nat_nat(div_mod_nat(Q_2),C_1))),R_3)))))))) # label(fact_4211_divmod__nat__rel__mult2__eq) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3772 (all P_3 all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),B_2)) -> -(hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),A_3)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),B_2)) & hBOOL(hAPP_nat_bool(prime,P_3))))) # label(fact_4176_coprime__prime) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3773 (all K_2 all L_2 hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(number_number_of_int,K_2)),hAPP_int_int(number_number_of_int,L_2)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,hAPP_int_rat(number_number_of_rat,K_2)),hAPP_int_rat(number_number_of_rat,L_2))) # label(fact_4709_Fract__number__of__quotient) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3774 (all B_1_1 all B_2_1 is_bool(hAPP_f115998247l_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__fun_Itc__prod_Itc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_Mtc__p) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3775 (all A_215 all M_24 all N_64 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_24),N_64)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_215),M_24)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_215),N_64))))) # label(fact_554_le__imp__power__dvd) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3776 (all N_57 all A_211 all B_157 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_211),B_157)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_211)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_57)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_211),N_57)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_157),N_57))))))) # label(fact_605_power__strict__mono) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3777 (all A_228 A_228 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_228),one_one_nat)) # label(fact_328_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3778 (all X_1 all Y_1 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_1),hAPP_complex_complex(invers1449016382omplex,Y_1)) = hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,X_1),Y_1)) # label(fact_3971_complex__divide__def) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3779 (all A_196 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_196),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),zero_zero_int))) # label(fact_795_power2__less__0) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3780 (all C_76 all B_133 all A_157 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_133),A_157)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_76),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_157),C_76)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_133),C_76)))))) # label(fact_1197_mult__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3781 (all M_6 all N_20 hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_6),N_20)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(semiri1621563631at_int,M_6)),N_20)) # label(fact_2985_of__nat__power) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3782 (all I_1 all K_1 (is_int(K_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),I_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),K_1)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),zero_zero_int)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),I_1)) | K_1 = zero_zero_int <-> zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,I_1),K_1)))) # label(fact_2634_zdiv__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3783 (all A_170 all B_145 all K_7 (hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_145),K_7) = A_170 -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,B_145),A_170)))) # label(fact_1100_dvdI) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3784 (all X_48 all N_69 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_48),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_69)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_48),N_69)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_48),N_69))) # label(fact_45_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3785 (all X_32 one_one_complex = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_32),zero_zero_nat)) # label(fact_952_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3786 (all M all N_1 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) -> hAPP_nat_nat(div_mod_nat(M),N_1) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)),N_1))) # label(fact_2514_mod__geq) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3787 (all C all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),A)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),A)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C),zero_zero_int)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C)))) # label(fact_1323_mult__less__cancel__left__disj) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3788 (all I (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),I)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_nat(nat,I))),one_one_nat) = hAPP_list_int_nat(size_size_list_int,quickc666637781d_zero(I)))) # label(fact_4288_length__around__zero) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3789 (all N_1 hAPP_real_real(hAPP_n546711566l_real(root,N_1),zero_zero_real) = zero_zero_real) # label(fact_3894_real__root__zero) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3790 (all C_66 all D_13 all A_132 all B_111 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_132),B_111)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_66),D_13)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_111)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_66)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_132),C_66)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_111),D_13)))))))) # label(fact_1407_mult__strict__mono) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3791 (all B all A all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(wset(A),P_5)))))))) # label(fact_2241_wset__mem) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3792 (all K_1 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),K_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),N)))) # label(fact_1772_dvd__reduce) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3793 (all C_37 all A_94 all B_77 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_94),B_77)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_94),C_37)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_77),C_37))))) # label(fact_2065_add__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3794 (all J_2 all K_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),J_2)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),K_2)),M)) -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),J_2) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),K_2)))) # label(fact_1503_neg__one__power__eq__mod__m) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3795 (all A_133 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_133),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_133),one_one_int)))) # label(fact_1368_less__add__one) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3796 (all A_46 all B_39 all C_14 hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_46),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_39),C_14))),B_39) = hAPP_C498520661umeral(div_mo1740067990umeral(A_46),B_39)) # label(fact_2421_mod__mult__self2) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3797 (all F all A all B hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,A)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,B)),hAPP_real_real(F,A))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B),A))),A)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,B)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,B)),hAPP_real_real(F,A))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B),A))),B))) # label(fact_3371_lemma__MVT) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3798 (all N_1 (zero_zero_nat != N_1 -> (exists M_1 N_1 = hAPP_nat_nat(suc,M_1)))) # label(fact_3308_not0__implies__Suc) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3799 (all X_2 hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),cos)),cos),X_2),hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(cos,X_2))),hAPP_real_real(sin,X_2)))))) # label(fact_4791_DERIV__cos__cos__mult3) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3800 (all N_1 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,N_1))),zero_zero_int))) # label(fact_3539_negative__zle__0) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3801 (all J all K_1 all I_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K_1)),I_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1))))) # label(fact_1678_le__diff__conv) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3802 (all V_16 all W_14 all Z_14 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_16)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,W_14)),Z_14)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_16),W_14))),Z_14)) # label(fact_421_mult__number__of__left) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3803 (all X_1 all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_nat_real(real_nat,A_3))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X_1)),A_3)))) # label(fact_3675_natceiling__le) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3804 (all N_1 (zero_zero_nat != N_1 -> hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1))) # label(fact_3233_Suc__Suc__mult__two__diff__two) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3805 (all Y_2 all X_2 (hBOOL(hAPP_int_bool(nat_is_nat,X_2)) -> hBOOL(hAPP_f448129468l_bool(nat_nat_set,hAPP_i1948725293t_bool(ord_at875362053st_int(X_2),Y_2))))) # label(fact_5143_SetInterval_Otransfer__int__nat__set__function__closures) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3806 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> hAPP_real_real(tan,X_1) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1))),one_one_real))))) # label(fact_3629_tan__half) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3807 (all A_3 all B_2 hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)))) # label(fact_5005_Fract__coprime) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3808 (all X_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),one_one_real)),hAPP_nat_real(real_nat,natfloor(X_1))))) # label(fact_3692_real__natfloor__gt__diff__one) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3809 (all Nat hAPP_nat_nat(size_size_nat,hAPP_nat_nat(suc,Nat)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(size_size_nat,Nat)),hAPP_nat_nat(suc,zero_zero_nat))) # label(fact_4056_nat_Osize_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3810 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M)) -> N_1 = M))) # label(fact_1724_le__antisym) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3811 (all N all Ma (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,N),Ma)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,quickcheck_int_of(N)),quickcheck_int_of(Ma))))) # label(fact_4251_less__eq__code__int__def) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3812 (all Ta all D all D_1 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,D),D_1)) -> (all X all K (-hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,D),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,K),D_1))),Ta))) <-> -hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,D),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X),Ta))))))) # label(fact_1818_inf__period_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3813 (all A_27 all B_24 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_27),B_24)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_27),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B_24),A_27)) = B_24)) # label(fact_2592_dvd__mult__div__cancel) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3814 (all D_3 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_3),hAPP_int_int(abs_abs_int,D_3)))) # label(fact_3419_zdvd__self__abs1) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3815 (all P_1 all N ((all M_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N)) -> hBOOL(hAPP_nat_bool(P_1,M_1)))) <-> (all X (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),hAPP_n1699378549t_bool(ord_at238088361st_nat(zero_zero_nat),N))) -> hBOOL(hAPP_nat_bool(P_1,X)))))) # label(fact_5069_all__nat__less) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3816 (all W_1 all Z_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_1),one_one_int)),Z_2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2)))) # label(fact_132_add1__zle__eq) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3817 (all Z_1 all X_1 all D_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),Z_1))),one_one_int)),D_3)))))) # label(fact_3464_incr__lemma) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3818 (all A_70 all B_62 all C_30 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(div_mod_int(A_70),C_30)),hAPP_int_int(div_mod_int(B_62),C_30))),C_30) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_70),B_62)),C_30)) # label(fact_2322_mod__mult__eq) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3819 (all P all Q all R hAPP_P2110489235nt_int(hAPP_f1855025978nt_int(hAPP_f1038672605nt_int(cOMBB_1506897658nt_int,P),Q),R) = hAPP_int_fun_int_int(P,hAPP_P1175774780nt_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_012) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3820 (all B_2 all A_3 all X_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),X_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),X_1)))) # label(fact_4324_gcd__semilattice__nat_Ole__infI1) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3821 (all C_63 all D_10 all A_129 all B_108 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_129),B_108)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,C_63),D_10)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_129)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_63)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_129),C_63)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_108),D_10)))))))) # label(fact_1428_mult__le__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3822 (all A_23 all B_20 zero_z891286103de_int = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_23),B_20)),B_20)) # label(fact_2616_mod__div__trivial) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3823 (all A all B (is_int(A) & is_int(B) -> (zero_zero_int = A | zero_zero_int = B <-> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),B) = zero_zero_int))) # label(fact_1032_mult__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3824 (all A_193 all M_21 all N_45 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_193),M_21)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_193),N_45)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_193),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_21),N_45))) # label(fact_822_power__add) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3825 (all B all A (zero_zero_rat = A <-> hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B),A) = B)) # label(fact_640_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3826 (all W (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W))) -> hAPP_P1175774780nt_int(product_fst_int_int,posDivAlg(one_one_int,hAPP_int_int(number_number_of_int,W))) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,one_one_int),hAPP_int_int(number_number_of_int,W)))) # label(fact_4226_div__pos__pos__1__number__of) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3827 (all P_1 ((exists X hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,X)))) <-> (exists X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) & hBOOL(hAPP_int_bool(P_1,X)) & is_int(X))))) # label(fact_2988_transfer__int__nat__quantifiers_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3828 (all W_16 hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit1,pls)),W_16)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,one_one_rat),hAPP_int_rat(number_number_of_rat,W_16))) # label(fact_53_add__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3829 (all P all Q all R hAPP_r1562208522t_bool(hAPP_f2084827628t_bool(hAPP_f1121609875t_bool(cOMBB_2125215742l_real,P),Q),R) = hAPP_f1902329768t_bool(P,hAPP_r285062775l_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Nat__Onat_Mtc__fun_) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3830 (all X_2 all P_5 (zero_zero_int != hAPP_int_int(standardRes(P_5),X_2) <-> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_2),zero_zero_int),P_5)))) # label(fact_2447_StandardRes__prop3) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3831 (all X_2 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) -> one_one_int = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_nat(nat,X_2))))) # label(fact_2808_EvenOdd_Oneg__one__even__power) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3832 (all C_71 all B_123 all A_146 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_123),A_146)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_71),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_146),C_71)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_123),C_71)))))) # label(fact_1257_mult__strict__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3833 (all A_109 all B_88 all C_48 (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_109),C_48) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_109),B_88) -> B_88 = C_48)) # label(fact_1927_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3834 (all X_2 all Y_2 all B (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),B)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B),X_2)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B),Y_2))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),Y_2))))) # label(fact_537_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3835 (all Y_1 all X_1 ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_1),X_1)) -> hAPP_int_int(nat_tsub(X_1),Y_1) = zero_zero_int) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_1),X_1)) -> hAPP_int_int(nat_tsub(X_1),Y_1) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),Y_1)))) # label(fact_2802_tsub__def) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3836 (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2)) -> hBOOL(hAPP_real_bool(isCont_real_real(ln),X_2)))) # label(fact_5157_isCont__ln) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3837 (all X_1 all K_2 all N_1 all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (zero_zero_nat != N_1 -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),N_1) -> (exists I_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),I_2) = X_1))))) # label(fact_4203_prime__power__exp) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3838 (all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,P_5),zOdd)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5))))) # label(fact_2704_zprime__zOdd__eq__grt__2) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3839 (all C all A all B (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A),B)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A),C)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,B),C)) <-> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,B),A)),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,C),A))))))) # label(fact_2549_div__dvd__div) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3840 (all Ma all K_1 all N (N = Ma <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),K_1))) # label(fact_1719_nat__add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3841 (all F all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) -> ((all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),B)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X)) -> (exists Y (hBOOL(hAPP_real_bool(deriv_real(F,X),Y)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y)))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(F,A)),hAPP_real_real(F,B)))))) # label(fact_4549_DERIV__nonneg__imp__nonincreasing) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3842 (all A_181 hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_181),A_181))) # label(fact_982_dvd__refl) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3843 (all W_7 hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,W_7)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,one_one_complex),one_one_complex)),hAPP_int_complex(number528085621omplex,W_7))) # label(fact_477_double__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3844 (all P all Q all R hAPP_r794344869l_real(hAPP_f517738447l_real(hAPP_f1879982819l_real(cOMBB_731595089l_real,P),Q),R) = hAPP_f1607917947l_real(P,hAPP_r1134773055l_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__HOL__Obool_J_000tc__fun_043) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3845 (all N_1 (N_1 != zero_zero_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1)),one_one_nat)))) # label(fact_3245_Suc__mult__two__diff__one) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3846 (all P all Q all R hAPP_rat_rat(P,hAPP_nat_rat(Q,R)) = hAPP_nat_rat(hAPP_f2062659861at_rat(hAPP_f1082758869at_rat(cOMBB_rat_rat_nat,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Rat__Orat_000tc__Rat__Orat_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3847 (all C_1 all A_3 all B_2 all Q_2 all R_3 (is_int(B_2) -> (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3))) -> (zero_zero_int != B_2 -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1)) -> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),C_1)),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,Q_2),C_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(div_mod_int(Q_2),C_1))),R_3))))))))) # label(fact_2783_zmult2__lemma) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3848 (all N ((exists M_1 N = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),M_1))) <-> -hBOOL(hAPP_nat_bool(even_odd_even_nat,N)))) # label(fact_3786_odd__Suc__mult__two__ex) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3849 (all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N_1) = hAPP_nat_nat(suc,N_1)) # label(fact_3073_Suc__eq__plus1__left) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3850 (all A_88 A_88 = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,one_on1684967323de_int),A_88)) # label(fact_2126_mult__1__left) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3851 (all B_55 all A_63 hAPP_int_int(div_mod_int(A_63),B_55) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_55),A_63)),B_55)) # label(fact_2346_mod__add__self1) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3852 (all X_2 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Y_2)) <-> X_2 = Y_2 | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Y_2)))) # label(fact_1513_less__eq__real__def) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3853 (all B_117 all A_140 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_117),A_140))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_140)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_117))))) # label(fact_1290_zero__less__mult__pos2) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3854 (all A_74 hAPP_C498520661umeral(div_mo1740067990umeral(A_74),zero_z126310315umeral) = A_74) # label(fact_2308_mod__by__0) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3855 (all X_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,X_2)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),pls)))) # label(fact_765_le__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3856 (all A_3 all B_2 all X_1 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),B_2)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),X_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),X_1)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2))))) # label(fact_4312_gcd__semilattice__nat_Oless__infI2) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3857 (all X_2 (X_2 = zero_zero_complex <-> X_2 = zero_zero_complex)) # label(fact_1900_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3858 (all X_10 all Y_7 ((all Z (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z),X_10)),Y_7))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_10),Y_7)))) # label(fact_2817_field__le__mult__one__interval) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3859 (all Ta all A_1 all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ta),A_1)) -> (all X (is_int(X) -> ((all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A_1)) -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) != X)))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),Ta)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),D_1)),Ta))))))))) # label(fact_5096_aset_I5_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3860 (all P all Q all R hAPP_r1250527377l_real(hAPP_r337325687l_real(hAPP_f1644353557l_real(cOMBC_1508019322l_real,P),Q),R) = hAPP_r1250527377l_real(hAPP_r337325687l_real(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__fun_Itc_) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3861 (all R_3 all A_3 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_real_real(cos,A_3)) = hAPP_complex_real(re,rcis(R_3,A_3))) # label(fact_4261_Re__rcis) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3862 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(ln,hAPP_real_real(exp_real,one_one_real))),hAPP_real_real(ln,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_real_real(ln,X_1)) = hAPP_real_real(log(hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X_1))) # label(fact_3859_log__base__10__eq1) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3863 (all N_1 hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_1)))))) # label(fact_3546_neg__zminus__int) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3864 (all A_3 all B_2 hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(uminus_uminus_int,A_3)),hAPP_int_int(uminus_uminus_int,B_2))) # label(fact_3522_zdiv__zminus__zminus) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3865 (all N_1 (is_int(N_1) -> N_1 = archim1246769320r_real(hAPP_int_real(real_int,N_1)))) # label(fact_4379_floor__real__of__int) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3866 (all A_3 all B_2 all C_1 all D_3 hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)),hAPP_int_rat(hAPP_int_fun_int_rat(fract,C_1),D_3)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),D_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),C_1))) # label(fact_4687_divide__rat) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3867 (all C all D all A all B (D != C & A != B <-> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),C)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),D)) != hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),D)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),C)))) # label(fact_317_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3868 (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_real_real(exp_real,X_2))))) # label(fact_3839_one__le__exp__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3869 (all A_201 all N_50 all N_49 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_50),N_49)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_201)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_201),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_201),N_49)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_201),N_50))))))) # label(fact_683_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3870 (all C all A all B (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A),B)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A),C)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,B),A)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,C),A))) <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,B),C)))))) # label(fact_2550_div__dvd__div) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3871 (all X_1 all Y_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_1),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))),one_one_real))) # label(fact_3379_cos__x__y__le__one) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3872 (all Z_4 all Z_1 all W_15 all W (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_15),W)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_4),Z_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_15),Z_4)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W),Z_1)))))) # label(fact_96_zadd__zless__mono) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3873 (all C_78 all D_14 all A_159 all B_135 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_159),B_135)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_78),D_14)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_159)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_78)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_159),C_78)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_135),D_14)))))))) # label(fact_1190_mult__mono_H) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3874 (all A_43 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_43),zero_zero_complex) = zero_zero_complex) # label(fact_2475_mult_Ozero__right) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3875 (all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1)) -> (exists X hAPP_real_real(exp_real,X) = Y_1))) # label(fact_3942_exp__total) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3876 (all P all Q all R hAPP_P603027463t_bool(hAPP_i119638232t_bool(hAPP_f2018027565t_bool(cOMBC_33652895t_bool,P),Q),R) = hAPP_int_bool(hAPP_P252494982t_bool(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oin_003) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3877 (all X_2 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (X_2 = zero_zero_real <-> hAPP_real_real(hAPP_n546711566l_real(root,N),X_2) = zero_zero_real))) # label(fact_3903_real__root__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3878 (all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(div_mod_nat(Ma),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) <-> one_one_nat = hAPP_nat_nat(div_mod_nat(Ma),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_2525_mod2__gr__0) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3879 (all B_90 all A_111 all C_50 (hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_50),A_111) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_90),A_111) -> C_50 = B_90)) # label(fact_1910_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3880 (all A_3 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natceiling(X_1)),A_3) = natceiling(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_nat_real(real_nat,A_3))))) # label(fact_3705_natceiling__add) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3881 (all Ma all K_1 all F ((all M_1 all N_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,M_1)),hAPP_nat_nat(F,N_2))))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(F,Ma)),K_1)),hAPP_nat_nat(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),K_1)))))) # label(fact_2235_mono__nat__linear__lb) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3882 (all P_1 all K_1 all I_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),I_1)) -> (hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_1),one_one_int))) -> ((all I_2 (is_int(I_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),I_2)) -> (hBOOL(hAPP_int_bool(P_1,I_2)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I_2),one_one_int))))))) -> hBOOL(hAPP_int_bool(P_1,I_1)))))) # label(fact_2234_int__gr__induct) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3883 (all A_108 all B_87 all C_47 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_108),B_87)),C_47) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_108),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_87),C_47))) # label(fact_1946_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3884 (all A_55 all B_47 all C_19 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_55),B_47)),C_19) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_55),hAPP_int_int(div_mod_int(B_47),C_19))),C_19)) # label(fact_2374_mod__diff__right__eq) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3885 (all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(arccos,Y_1))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arccos,Y_1)),pi)) & hAPP_real_real(cos,hAPP_real_real(arccos,Y_1)) = Y_1))) # label(fact_3623_arccos) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3886 (all X_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),natfloor(X_1)))) # label(fact_3384_zero__le__natfloor) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3887 (all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N)) -> hAPP_nat_int(semiri1621563631at_int,hAPP_f957591787ol_nat(finite_card_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),N))))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,N),one_one_int))) # label(fact_4774_int__card__bdd__int__set__l__l) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3888 (all X_48 all N_69 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_48),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_69)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_48),N_69)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_48),N_69))) # label(fact_48_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3889 (all A_3 all B_2 ((zero_zero_int = hAPP_int_int(div_mod_int(A_3),B_2) -> zero_zero_int = hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,A_3)),B_2)) & (zero_zero_int != hAPP_int_int(div_mod_int(A_3),B_2) -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_2),hAPP_int_int(div_mod_int(A_3),B_2)) = hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,A_3)),B_2)))) # label(fact_3544_zmod__zminus1__eq__if) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3890 (all M_6 all N_20 hAPP_nat_rat(semiri151668891at_rat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_6),N_20)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_nat_rat(semiri151668891at_rat,M_6)),N_20)) # label(fact_2981_of__nat__power) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3891 (all F all J all N all H (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H)) -> (exists B_6 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),J)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,H))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_6),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),N)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N))))) = hAPP_real_real(F,H)))) # label(fact_4927_Maclaurin__lemma) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3892 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)))) # label(fact_2217_less__iff__diff__less__0) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3893 (all J all I_1 all P_1 all P_5 ((all I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),P_5)) -> (hBOOL(hAPP_nat_bool(P_1,I_2)) -> hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,I_2)),P_5)))))) -> (hBOOL(hAPP_nat_bool(P_1,I_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),P_5)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J),P_5)) -> hBOOL(hAPP_nat_bool(P_1,J))))))) # label(fact_3251_mod__induct) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3894 (all N all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,N)),hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,Ma)))) <-> zero_zero_nat = N & Ma = zero_zero_nat)) # label(fact_3538_int__zle__neg) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3895 (all R_3 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,R_3),one_one_real)),hAPP_int_real(real_int,archim1246769320r_real(R_3))))) # label(fact_4455_real__of__int__floor__gt__diff__one) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3896 (all N all Ma hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(suc,N)),Ma) = hAPP_nat_nat(nat_case_nat(hAPP_nat_nat(suc,N),hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,suc),hAPP_nat_fun_nat_nat(ord_max_nat,N))),Ma)) # label(fact_5192_max__Suc1) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3897 (all Y_6 all P_6 all N_10 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_6),N_10)) -> hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N_10)),P_6)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),P_6))),Y_6))) # label(fact_3178_lemma__realpow__diff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3898 (all V_13 all W_11 all Z_13 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_13),W_11))),Z_13) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_13)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,W_11)),Z_13))) # label(fact_433_add__number__of__left) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3899 (all N_19 zero_zero_real = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),hAPP_nat_nat(suc,N_19))) # label(fact_2997_power__0__Suc) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3900 (all C_1 all D_3 all A_3 all B_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C_1),D_3),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),C_1)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_2),D_3)),M))))) # label(fact_1537_zcong__zdiff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3901 (all A_38 all A_37 all B_30 hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_38),B_30)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_37),B_30)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_38),A_37)),B_30)) # label(fact_2491_mult_Odiff__left) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3902 (all P all Q all R hAPP_f1406902706l_real(P,hAPP_n1699378549t_bool(Q,R)) = hAPP_nat_real(hAPP_f292751352t_real(hAPP_f1607377819t_real(cOMBB_1533963633al_nat,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__RealDef_) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3903 (all A all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),zero_zero_int),P_5)) -> comple219730294t_bool(setS(A,P_5)) = hAPP_i1948725293t_bool(sRStar,P_5))))) # label(fact_3306_MultInvPair__prop2) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3904 (all M_13 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,one_one_rat),hAPP_nat_rat(semiri151668891at_rat,M_13)) = hAPP_nat_rat(semiri151668891at_rat,hAPP_nat_nat(suc,M_13))) # label(fact_2846_of__nat__Suc) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3905 (all X_36 all Y_28 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_36),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_28),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Y_28)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_36),Y_28))))) # label(fact_844_power2__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3906 (all M all K_2 all N_1 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,K_2),N_1) = one_one_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N_1))) # label(fact_4355_gcd__mult__cancel__nat) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3907 (all P all Q all R hAPP_r1250527377l_real(hAPP_n682674106l_real(P,R),hAPP_nat_real(Q,R)) = hAPP_n546711566l_real(hAPP_f451358433l_real(hAPP_f335634176l_real(cOMBS_337378985l_real,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__fun_Itc__Real) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3908 (all M all N_1 hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N_1)),hAPP_nat_nat(div_mod_nat(M),N_1)) = divmod_nat(M,N_1)) # label(fact_4210_divmod__nat__div__mod) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3909 (all X_1 (zero_zero_real != hAPP_real_real(cos,X_1) -> (hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1)) != zero_zero_real -> hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(tan,X_1))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(tan,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1))))) # label(fact_3593_tan__double) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3910 (all A_1 all B_1 (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,image_nat_int(semiri1621563631at_int,A_1)),image_nat_int(semiri1621563631at_int,B_1))) <-> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A_1),B_1)))) # label(fact_4600_transfer__nat__int__set__relations_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3911 (all X_1 all Y_1 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_1)),Y_1))) # label(fact_4960_gcd__dvd2__int) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3912 (all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,pls)),Y_2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(number_number_of_int,Y_2))))) # label(fact_287_less__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3913 (all B_73 all A_84 all C_36 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_36)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_73),A_84)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_73),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_84),C_36)))))) # label(fact_2159_add__increasing2) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3914 (all M all N_1 all K_2 ((K_2 = zero_zero_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_1)) = zero_zero_nat) & (K_2 != zero_zero_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_1))))) # label(fact_2722_nat__mult__div__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3915 (all A_3 all B_2 (is_int(A_3) -> A_3 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2))),hAPP_int_int(div_mod_int(A_3),B_2)))) # label(fact_2637_zmod__zdiv__equality) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3916 (all A_64 all B_56 hAPP_C498520661umeral(div_mo1740067990umeral(A_64),B_56) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_64),B_56)),B_56)) # label(fact_2344_mod__add__self2) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3917 (all A_106 (is_int(A_106) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_106),zero_zero_int) = A_106)) # label(fact_1971_add__0__right) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3918 (all A_166 all B_141 hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_166),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_141),A_166)))) # label(fact_1130_dvd__triv__right) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3919 (all P all Q all R hAPP_n546711566l_real(hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,P),Q),R) = hAPP_r1250527377l_real(P,hAPP_nat_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__R_008) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3920 (all Nat_3 hAPP_nat_nat(suc,Nat_3) != zero_zero_nat) # label(fact_2852_nat_Osimps_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3921 (all Z_8 all X_25 all Y_19 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_25),Y_19)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_25),Z_8)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_25),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Y_19),Z_8)))))) # label(fact_1531_dvd__diff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3922 (all X_1 all Y_1 X_1 = hAPP_complex_real(re,hAPP_real_complex(hAPP_r265291036omplex(complex,X_1),Y_1))) # label(fact_4112_Re) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3923 (all A_8 all N_14 all B_7 (hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_8),hAPP_nat_nat(suc,N_14)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_7),hAPP_nat_nat(suc,N_14)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_8)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_7)) -> B_7 = A_8)))) # label(fact_3115_power__inject__base) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3924 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,X_1)),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),one_one_real)))) # label(fact_3403_less__one__imp__sqr__less__one) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3925 (all X_1 all Y_1 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_1))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X_1)),hAPP_real_real(cos,Y_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_1)),hAPP_real_real(sin,Y_1))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_1))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_1)),hAPP_real_real(cos,Y_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X_1)),hAPP_real_real(sin,Y_1))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_real) # label(fact_3740_sin__cos__add) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3926 (all L all U hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(ord_gr1297742076an_int(L),U)) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,U),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,L),one_one_int)))) # label(fact_4627_card__greaterThanLessThan__int) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3927 (all Z_1 all X_1 all Y_1 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),Y_1),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),Z_1)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_1),Z_1)),M)))) # label(fact_1168_zcong__zpower) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3928 (all X_1 hAPP_complex_real(re,X_1) = hAPP_complex_real(re,hAPP_complex_complex(cnj,X_1))) # label(fact_4114_complex__Re__cnj) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3929 (all V_22 all V_21 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_21)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_22)) -> hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_21),V_22)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_21)),hAPP_int_rat(number_number_of_rat,V_22))))) # label(fact_371_semiring__mult__number__of) # label(axiom) # label(non_clause). [assumption]. 5.69/5.71 3930 (all A_25 all B_22 (is_int(B_22) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_25),B_22)) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_22),A_25)),A_25) = B_22))) # label(fact_2599_dvd__div__mult__self) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3931 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(suc,N_1))) -> (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) -> M = hAPP_nat_nat(suc,N_1)))) # label(fact_2872_le__SucE) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3932 (all P (is_bool(P) -> fTrue = P | P = fFalse)) # label(help_If_3_1_If_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_T) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3933 (all N_19 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,zero_zero_rat),hAPP_nat_nat(suc,N_19)) = zero_zero_rat) # label(fact_2993_power__0__Suc) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3934 (all P_1 all N all K_1 (is_int(K_1) & is_int(N) -> ((zero_zero_int = K_1 -> hBOOL(hAPP_int_bool(P_1,zero_zero_int))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_1)) -> (all I_2 all J_1 (is_int(J_1) & is_int(I_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),J_1)) & hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),I_2)),J_1) = N & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,J_1),K_1)) -> hBOOL(hAPP_int_bool(P_1,I_2)))))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),zero_zero_int)) -> (all I_2 all J_1 (is_int(I_2) & is_int(J_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),J_1)) & hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),I_2)),J_1) = N & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J_1),zero_zero_int)) -> hBOOL(hAPP_int_bool(P_1,I_2)))))) <-> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N),K_1)))))) # label(fact_2693_split__zdiv) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3935 (all X_1 all Y_1 hAPP_P1975530577nt_int(hAPP_P1145851913nt_int(hAPP_b40753821nt_int(if_Pro1731782967nt_int,fTrue),X_1),Y_1) = X_1) # label(help_If_1_1_If_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_T) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3936 (all A_184 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_184),zero_zero_nat))) # label(fact_917_dvd__0__right) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3937 (all X_2 all P_5 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),hAPP_i1948725293t_bool(sRStar,P_5))) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),hAPP_i1948725293t_bool(sr,P_5))))) # label(fact_2518_SRStar__SR__prop) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3938 (all X_29 all Y_23 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_29),Y_23)) -> (Y_23 != X_29 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_29),Y_23))))) # label(fact_1012_order__le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3939 (all C_89 all D_17 all A_200 all B_154 all R_8 (zero_zero_nat != R_8 -> (C_89 != D_17 & A_200 = B_154 -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_200),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,R_8),C_89)) != hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_154),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,R_8),D_17))))) # label(fact_694_add__scale__eq__noteq) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3940 (all A_234 all C_94 all D_19 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_234),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_94),D_19)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_234),C_94)),D_19)) # label(fact_226_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3941 (all M_26 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),one_one_real)),M_26) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,M_26),M_26)) # label(fact_408_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3942 (all B_128 all A_152 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_152),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_128)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_152),B_128)),zero_zero_int))))) # label(fact_1221_mult__nonpos__nonneg) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3943 (all B_118 all A_141 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_141),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_118)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_141),B_118)),zero_zero_int))))) # label(fact_1281_mult__neg__pos) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3944 (all B_97 all A_118 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_118),zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_97),zero_zero_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_118),B_97)),zero_zero_nat))))) # label(fact_1858_add__nonpos__neg) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3945 (all K_2 hAPP_nat_rat(semiri151668891at_rat,K_2) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_nat_int(semiri1621563631at_int,K_2)),one_one_int)) # label(fact_4676_Fract__of__nat__eq) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3946 (all P hAPP_nat_nat(cOMBI_nat,P) = P) # label(help_COMBI_1_1_COMBI_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3947 (all P all Q all R hAPP_f713949180nt_int(P,hAPP_i1153444985nt_int(Q,R)) = hAPP_i1836591008nt_int(hAPP_f617310012nt_int(hAPP_f197974549nt_int(cOMBB_638531587nt_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__prod_Itc__Int__Oint) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3948 (all N_56 all A_206 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_206)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_206),N_56))))) # label(fact_652_zero__le__power) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3949 (all A_235 all B_164 all C_95 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_235),B_164)),C_95) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_235),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_164),C_95))) # label(fact_224_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3950 (all A all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),zero_zero_int),P_5)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_5),A)) -> (all X (hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X),setS(A,P_5))) -> hAPP_f957591787ol_nat(finite_card_int,X) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))))) # label(fact_3290_SetS__elems__card) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3951 (all P_1 all L ((exists X (hBOOL(hAPP_rat_bool(P_1,X)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,L),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X),zero_zero_rat))))) <-> (exists X hBOOL(hAPP_rat_bool(P_1,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,L),X)))))) # label(fact_1810_unity__coeff__ex) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3952 (all B_127 all A_151 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_151)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_127),zero_zero_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_127),A_151)),zero_zero_nat))))) # label(fact_1228_mult__nonneg__nonpos2) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3953 (all A_194 all B_153 all N_46 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_194),N_46)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_153),N_46)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_194),B_153)),N_46)) # label(fact_813_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3954 (all A_225 all M_28 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_225),M_28)),M_28) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_225),one_one_rat)),M_28)) # label(fact_398_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3955 (all F all X_2 all L (hBOOL(hAPP_real_bool(deriv_real(F,X_2),L)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,L),zero_zero_real)) -> (exists D_2 ((all H_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H_1),D_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),H_1))),hAPP_real_real(F,X_2)))))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2))))))) # label(fact_3889_DERIV__neg__dec__right) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3956 (all K_2 ((K_2 != zero_z126310315umeral -> (hAPP_C498520661umeral(div_mo1740067990umeral(K_2),hAPP_i769753017umeral(number1443263063umeral,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_z126310315umeral -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),code_int_of(hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,K_2),hAPP_i769753017umeral(number1443263063umeral,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = code_int_of(K_2)) & (zero_z126310315umeral != hAPP_C498520661umeral(div_mo1740067990umeral(K_2),hAPP_i769753017umeral(number1443263063umeral,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) -> code_int_of(K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),code_int_of(hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,K_2),hAPP_i769753017umeral(number1443263063umeral,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),one_one_int))) & (zero_z126310315umeral = K_2 -> code_int_of(K_2) = zero_zero_int))) # label(fact_2807_Code__Numeral_Oint__of__code) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3957 (all P_1 all A0 all A1 (hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(posDivAlg_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A0),A1))) -> ((all A_2 all B_4 (is_int(A_2) & is_int(B_4) -> (hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(posDivAlg_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_2),B_4))) -> ((-(hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),B_4)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_4),zero_zero_int))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_4)))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A_2),B_4)))))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A0),A1))))) # label(fact_2706_posDivAlg_Opinduct) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3958 (all K_2 all L_2 hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,hAPP_int_rat(number_number_of_rat,K_2)),hAPP_int_rat(number_number_of_rat,L_2)) = frct(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,K_2)),hAPP_int_int(number_number_of_int,L_2)))) # label(fact_4644_Frct__code__post_I6_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3959 (all X_1 all Y_1 X_1 = hAPP_P1860904029l_real(hAPP_P1982799375l_real(hAPP_b1482221219l_real(if_Pro313124157l_real,fTrue),X_1),Y_1)) # label(help_If_1_1_If_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_T) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3960 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natfloor(X_1)),one_one_nat) = natfloor(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),one_one_real)))) # label(fact_3404_natfloor__add__one) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3961 (all X_2 all Y_2 (zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) <-> X_2 = zero_zero_real & Y_2 = zero_zero_real)) # label(fact_1509_realpow__two__sum__zero__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3962 (all P all Q all R hAPP_f743790483t_bool(P,hAPP_r279757445t_bool(Q,R)) = hAPP_r2052264067t_bool(hAPP_f77155443t_bool(hAPP_f403082607t_bool(cOMBB_984071017l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_Mt) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3963 (all A_104 A_104 = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,zero_zero_real),A_104)) # label(fact_1990_add__0__left) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3964 (all A_87 all B_76 A_87 = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_87),B_76)),B_76)) # label(fact_2134_add__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3965 (all Ta all A_1 all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1)) -> (all X (is_int(X) -> ((all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A_1)) -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) != X)))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ta),X)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ta),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),D_1))))))))) # label(fact_5088_aset_I8_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3966 (all P all Q all R hAPP_r481142414t_bool(hAPP_f1625314454t_bool(hAPP_f340152631t_bool(cOMBB_1453147152l_real,P),Q),R) = hAPP_f2119767738t_bool(P,hAPP_r632163725t_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__HOL_) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3967 (all X_1 X_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,zero_zero_nat),X_1)) # label(fact_4337_gcd__0__left__nat) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3968 (all N all Ma (is_int(N) & is_int(Ma) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma)) -> (one_one_int = Ma & one_one_int = N <-> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ma),N) = one_one_int)))) # label(fact_754_pos__zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3969 (all A_1 all K_1 (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),hAPP_n1699378549t_bool(ord_at4362885an_nat(K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),hAPP_f22106695ol_nat(finite_card_nat,A_1))))) -> A_1 = hAPP_n1699378549t_bool(ord_at4362885an_nat(K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),hAPP_f22106695ol_nat(finite_card_nat,A_1))))) # label(fact_4908_subset__card__intvl__is__intvl) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3970 (all B_112 all C_67 all A_134 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_134)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_112),C_67)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_112),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_134),C_67)))))) # label(fact_1332_pos__add__strict) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3971 (all N_1 all X_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N_1),X_1))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,N_1)),hAPP_int_real(real_int,X_1))))) # label(fact_4414_real__of__int__div4) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3972 (all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N)) -> hAPP_f957591787ol_nat(finite_card_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),N)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat,N)),one_one_nat))) # label(fact_4775_card__bdd__int__set__l__l) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3973 (all M_17 all N_30 ((N_30 != zero_zero_nat -> hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,M_17),N_30) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,M_17),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,M_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_30),one_one_nat)))) & (zero_zero_nat = N_30 -> one_on1645066479umeral = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,M_17),N_30)))) # label(fact_1832_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3974 (all A_3 all B_2 (is_int(B_2) & is_int(A_3) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int)) -> (-(zero_zero_int = B_2 | A_3 = zero_zero_int) -> norm_frac(A_3,B_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(int_gcd,A_3),B_2))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),hAPP_int_int(hAPP_int_fun_int_int(int_gcd,A_3),B_2)))) & (B_2 = zero_zero_int | zero_zero_int = A_3 -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int) = norm_frac(A_3,B_2))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int)) -> norm_frac(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2)) = norm_frac(A_3,B_2)))) # label(fact_4806_norm__frac_Osimps) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3975 (all X_2 all N (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,N))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(number_number_of_nat,N)),natfloor(X_2))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,N)),X_2)))))) # label(fact_3409_le__natfloor__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3976 (all C_33 all D_5 all A_78 all B_67 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_78),B_67)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_33),D_5)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_78),C_33)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_67),D_5)))))) # label(fact_2209_add__less__le__mono) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3977 (all B_2 all X_1 all A_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),A_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),B_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)))))) # label(fact_4319_gcd__semilattice__nat_Ole__infI) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3978 (all A all E_2 all C all B all D (hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B)),E_2)),C) = D <-> hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),E_2)),D) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),E_2)),C))) # label(fact_1554_eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3979 (all X_1 all Y_1 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(real_int,X_1)),hAPP_int_real(real_int,Y_1)) = hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),Y_1))) # label(fact_4388_real__of__int__diff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3980 (all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> hAPP_nat_real(real_nat,hAPP_int_nat(nat,X_1)) = hAPP_int_real(real_int,X_1))) # label(fact_4448_real__nat__eq__real) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3981 (all Z_2 all P_1 ((all N_2 hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,N_2)))) -> ((all N_2 hBOOL(hAPP_int_bool(P_1,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_2)))))) -> hBOOL(hAPP_int_bool(P_1,Z_2))))) # label(fact_4547_int__of__nat__induct) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3982 (all N_48 all A_199 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_199)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_199),one_one_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_199),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_199),N_48))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_199),N_48)))))) # label(fact_721_power__Suc__less) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3983 (all A all N (A = zero_z126310315umeral & N != zero_zero_nat <-> zero_z126310315umeral = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A),N))) # label(fact_549_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3984 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int)) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) = hAPP_P1175774780nt_int(product_fst_int_int,hAPP_P1975530577nt_int(negateSnd,negDivAlg(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2))))))) # label(fact_4227_div__pos__neg) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3985 (all Y_1 all X_1 (hBOOL(hAPP_int_bool(nat_is_nat,X_1)) -> (hBOOL(hAPP_int_bool(nat_is_nat,Y_1)) -> hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Y_1)))))) # label(fact_5141_Nat__Transfer_Otransfer__int__nat__function__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3986 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N_1) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)),N_1))))) # label(fact_3186_div__geq) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3987 (all V (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V),pls)) <-> hAPP_int_nat(number_number_of_nat,V) = zero_zero_nat)) # label(fact_1000_eq__number__of__0) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3988 (all D all Sa exists Z all X (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X),Z)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X),Sa))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X),Sa)))))) # label(fact_2254_minf_I9_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3989 (all Z_2 all X_3 (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Z_2),X_3)) -> ((all X (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X),X_3)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Z_2)))) -> Z_2 = comple124823625p_real(X_3)))) # label(fact_4646_Sup__eq__maximum) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3990 (all V all W_1 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(number_number_of_nat,W_1)),hAPP_int_nat(number_number_of_nat,V))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_int_nat(number_number_of_nat,W_1))))) # label(fact_86_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3991 (all N_1 all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M)) -> hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),N_1))) # label(fact_3074_Suc__diff__le) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3992 (all A_208 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,zero_zero_real),A_208) = A_208) # label(fact_625_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3993 (all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),K_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),hAPP_int_int(bit0,K_1))))) # label(fact_965_rel__simps_I25_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3994 (all X_1 X_1 = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_1),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_1),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3366_real__sum__of__halves) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3995 (all P all Q all R hAPP_f203520653l_real(P,hAPP_r337325687l_real(Q,R)) = hAPP_r337325687l_real(hAPP_f1644353557l_real(hAPP_f1874145443l_real(cOMBB_1768400739l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__fun_Itc__RealDef__Oreal) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3996 (all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),P_3)))) # label(fact_4167_prime__g__zero) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3997 (all Ma all K_1 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),K_1))))) # label(fact_1801_mult__less__cancel2) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3998 (all R_3 all X_1 hAPP_complex_complex(cnj,hAPP_complex_complex(scaleR1652505878omplex(R_3),X_1)) = hAPP_complex_complex(scaleR1652505878omplex(R_3),hAPP_complex_complex(cnj,X_1))) # label(fact_5112_cnj_OscaleR) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 3999 (all A all A_1 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma)) -> (hBOOL(hAPP_int_bool(is_RRset(A_1),Ma)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),A_1)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hilbert_Eps_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,zcong(A)),Ma))),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),hAPP_i1948725293t_bool(norRRset,Ma))))),hAPP_i1948725293t_bool(norRRset,Ma))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),hilbert_Eps_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,zcong(A)),Ma))),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),hAPP_i1948725293t_bool(norRRset,Ma))))),Ma)))))) # label(fact_4892_aux__some) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4000 (all A all B ((all P_4 -(hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_4),B)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_4),A)) & hBOOL(hAPP_nat_bool(prime,P_4)))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A),B)))) # label(fact_4178_coprime__prime__eq) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4001 (all P_1 all L ((exists X (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,L),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),zero_zero_nat))) & hBOOL(hAPP_nat_bool(P_1,X)))) <-> (exists X hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,L),X)))))) # label(fact_1812_unity__coeff__ex) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4002 (all A_90 (is_int(A_90) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_90),one_one_int) = A_90)) # label(fact_2107_mult__1__right) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4003 (all Ma all N all Qr (Ma = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_P1654798560at_nat(product_fst_nat_nat,Qr)),N)),hAPP_P1654798560at_nat(product_snd_nat_nat,Qr)) & (N = zero_zero_nat -> hAPP_P1654798560at_nat(product_fst_nat_nat,Qr) = zero_zero_nat) & (zero_zero_nat != N -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_P1654798560at_nat(product_snd_nat_nat,Qr)),N)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_P1654798560at_nat(product_snd_nat_nat,Qr)))) & (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_P1654798560at_nat(product_snd_nat_nat,Qr))) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_P1654798560at_nat(product_snd_nat_nat,Qr)),zero_zero_nat)))) <-> hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(Ma,N),Qr)))) # label(fact_4240_divmod__nat__rel__def) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4004 (all Z_7 all W_2 all Y_10 all X_14 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Y_10),X_14)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_7),W_2)) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_14),W_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_10),Z_7)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_14),Y_10)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,W_2),Z_7))))) # label(fact_2607_div__mult__div__if__dvd) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4005 (all A all C all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),C)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),C))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C),zero_zero_real)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B),A)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C)))) # label(fact_1327_mult__less__cancel__right__disj) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4006 (all C_39 all D_7 all A_96 all B_79 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_96),B_79)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_39),D_7)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_96),C_39)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_79),D_7)))))) # label(fact_2052_add__strict__mono) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4007 (all P all Q all R hAPP_bool_bool(P,hAPP_int_bool(Q,R)) = hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_f627970963t_bool(cOMBB_bool_bool_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4008 (all N_44 hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,one_on1645066479umeral),N_44) = one_on1645066479umeral) # label(fact_828_power__one) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4009 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)),zero_zero_int)) -> hAPP_int_int(number_number_of_int,min) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)))) # label(fact_2699_div__pos__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4010 (all F all L all C (hBOOL(tendsto_real_real(F,L,at_real(C))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,L),zero_zero_real)) -> (exists R_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_1)) & (all X (X != C & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),X))),R_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,X)),zero_zero_real))))))))) # label(fact_5085_LIM__fun__less__zero) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4011 (all A_1 all X_2 (hBOOL(hAPP_int_bool(nat_is_nat,X_2)) -> (hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A_1)) <-> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_int_nat(nat,X_2)),image_int_nat(nat,A_1))))))) # label(fact_5147_transfer__int__nat__set__relations_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4012 (all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),hAPP_int_int(bit1,K_1))))) # label(fact_126_rel__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4013 (all N_1 zero_zero_real = hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),pi))) # label(fact_3640_sin__npi) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4014 (all N_1 one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N_1),one_one_int)),N_1)) # label(fact_4972_coprime__plus__one__int) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4015 (all N_35 all A_182 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_182)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_35)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_182),N_35)))))) # label(fact_977_one__less__power) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4016 (all W_6 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,zero_zero_complex),hAPP_int_complex(number528085621omplex,W_6))),hAPP_int_complex(number528085621omplex,W_6)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,W_6))) # label(fact_744_number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4017 (all X_2 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),one_one_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X_2)),one_one_real))))) # label(fact_3925_real__root__lt__1__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4018 (all F (hBOOL(summable_real(F)) -> ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(F,N_2)))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_f352196356l_real(suminf_real,F)))))) # label(fact_4857_suminf__gt__zero) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4019 (all A_174 all B_148 all C_86 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_174),C_86)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_148),C_86)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_174),B_148)),C_86)) # label(fact_1076_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4020 (all B_2 all A_3 all K_2 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2))),hAPP_int_int(div_mod_int(A_3),B_2))),K_2)) # label(fact_2639_zdiv__zmod__equality) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4021 (all P_1 all X_2 all Y_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_2)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2)) -> hBOOL(P_1)))) # label(fact_1713_dvd_Oless__imp__triv) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4022 (all B_1_1 is_bool(cauchy_real(B_1_1))) # label(gsy_c_SEQ_OCauchy_000tc__RealDef__Oreal) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4023 (all X_3 (hBOOL(summable_complex(X_3)) -> hBOOL(summable_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,re),X_3))))) # label(fact_4845_Re_Osummable) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4024 (all Z_9 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z_9),Z_9) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_9),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_502_mult__2__right) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4025 (all K_1 all Ma all N ((hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N))))) # label(fact_1825_mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4026 (all X_1 all Y_1 Y_1 = hAPP_P1860904029l_real(hAPP_P1982799375l_real(hAPP_b1482221219l_real(if_Pro313124157l_real,fFalse),X_1),Y_1)) # label(help_If_2_1_If_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_T) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4027 (all A_124 all C_58 all B_103 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_124),C_58)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_103),C_58))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_58)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_124),B_103))))) # label(fact_1459_mult__right__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4028 (all X_1 all Y_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Y_1)) -> bezw(X_1,Y_1) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_P1175774780nt_int(product_snd_int_int,bezw(Y_1,hAPP_nat_nat(div_mod_nat(X_1),Y_1)))),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_P1175774780nt_int(product_fst_int_int,bezw(Y_1,hAPP_nat_nat(div_mod_nat(X_1),Y_1)))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_snd_int_int,bezw(Y_1,hAPP_nat_nat(div_mod_nat(X_1),Y_1)))),hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X_1),Y_1))))))) # label(fact_4230_bezw__non__0) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4029 (all N_55 all A_205 all B_156 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_205),B_156)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_205)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_205),N_55)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_156),N_55)))))) # label(fact_659_power__mono) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4030 (all W_1 all Y_2 all X_2 all Z_2 (W_1 = X_2 | Z_2 = Y_2 <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,W_1),Z_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_2),Y_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,W_1),Y_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_2),Z_2)))) # label(fact_297_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4031 (all A_171 all B_146 all C_84 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_171),B_146)),C_84)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_171),C_84)))) # label(fact_1098_dvd__mult__left) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4032 (all X_2 hBOOL(hAPP_real_bool(deriv_real(sin,X_2),hAPP_real_real(cos,X_2)))) # label(fact_3861_DERIV__sin) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4033 (all Ma all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)) | hAPP_nat_nat(suc,N) = Ma <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),hAPP_nat_nat(suc,N))))) # label(fact_2875_le__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4034 (all Y_1 all X_1 (-hBOOL(hAPP_int_bool(even_odd_even_int,X_1)) -> (-hBOOL(hAPP_int_bool(even_odd_even_int,Y_1)) -> -hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_1)))))) # label(fact_2885_Parity_Oodd__times__odd) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4035 (all P all Q all R hAPP_complex_real(P,hAPP_nat_complex(Q,R)) = hAPP_nat_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Complex__Ocomplex_000tc__RealDef__Oreal_000tc__Nat__) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4036 (all Code_numeral_2 zero_z126310315umeral != code_S1047413653umeral(Code_numeral_2)) # label(fact_4081_code__numeral_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4037 (all X_1 all Y_1 hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,X_1),Y_1)) = hAPP_int_int(nat_tsub(hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_1))) # label(fact_3093_Nat__Transfer_Otransfer__int__nat__functions_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4038 (all N_1 all J_2 all K_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_2),K_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),N_1)),K_2)))) # label(fact_1641_less__imp__diff__less) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4039 (all N_1 all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M)) -> (M = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1) -> N_1 = hAPP_nat_nat(suc,zero_zero_nat)))) # label(fact_3135_mn__eq__m__one) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4040 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) -> zero_zero_nat = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(fact_fact_nat,N_1)),hAPP_nat_nat(fact_fact_nat,M)))) # label(fact_3813_fact__mod) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4041 (all N_60 all X_41 all Y_33 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,X_41),Y_33)) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_41),N_60)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_33),N_60))))) # label(fact_587_dvd__power__same) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4042 (all A_195 all N_47 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_195),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_195),N_47)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_195),N_47)),A_195)) # label(fact_804_power__commutes) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4043 (all A_165 all B_140 hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_165),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_165),B_140)))) # label(fact_1137_dvd__triv__left) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4044 (all A_3 cis(A_3) = rcis(one_one_real,A_3)) # label(fact_4259_cis__rcis__eq) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4045 (all Ma all N all K_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),N))))) # label(fact_1853_nat__mult__dvd__cancel1) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4046 (all X_2 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Y_2)),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_2),hAPP_real_real(uminus_uminus_real,X_2))))) # label(fact_3489_real__add__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4047 (all B_150 all A_178 (zero_z891286103de_int != A_178 -> (B_150 != zero_z891286103de_int -> zero_z891286103de_int != hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_178),B_150)))) # label(fact_1030_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4048 (all V_15 all W_13 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_15)),hAPP_int_real(number267125858f_real,W_13)) = hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_15),W_13))) # label(fact_426_arith__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4049 (all A_3 all B_2 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(ring_1_of_int_rat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2))),hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2))) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)),hAPP_int_rat(ring_1_of_int_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),one_one_int)))))) # label(fact_4725_rat__floor__lemma) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4050 (all X_2 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven)) -> zero_zero_int = hAPP_int_int(div_mod_int(X_2),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3234_even__div__2__prop1) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4051 (all X_2 all Y_2 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_i769753017umeral(number1443263063umeral,X_2)),hAPP_i769753017umeral(number1443263063umeral,Y_2))) <-> zero_z126310315umeral = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_i769753017umeral(number1443263063umeral,Y_2)),hAPP_i769753017umeral(number1443263063umeral,X_2)))) # label(fact_2441_dvd__eq__mod__eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4052 (all X_30 all N_34 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_34)) | one_one_int = X_30 -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_30),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_30),N_34))))) # label(fact_997_dvd__power) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4053 (all C_69 all A_144 all B_121 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_144),B_121)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_69)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_69),A_144)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_69),B_121)))))) # label(fact_1269_mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4054 (all Lx_3 all Ly_1 all Rx_3 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_3),Ly_1)),Rx_3) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_3),Rx_3)),Ly_1)) # label(fact_160_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4055 (all C_81 all A_167 all B_142 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_167),B_142)) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_167),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_142),C_81))))) # label(fact_1122_dvd__mult2) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4056 (all A all B all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X_2),A)),hAPP_real_real(hAPP_r1250527377l_real(powr,X_2),B))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B))))) # label(fact_4432_powr__le__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4057 (all B all C all A all D (hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A),D) = one_one_nat -> (one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,B),C) -> (B = A & D = C <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B),D) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A),C))))) # label(fact_4353_coprime__crossproduct__nat) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4058 (all A_3 all R_3 all B_2 all M all C_1 all D_3 all N_1 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_1),N_1))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_3),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,D_3),N_1)))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_3),B_2))),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_3),D_3))),N_1))) # label(fact_1567_xzgcda__linear__aux1) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4059 (all A all B all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(B),A),Ma)))) # label(fact_1002_zcong__sym) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4060 (all Y_2 all X_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2)) <-> Y_2 = X_2))) # label(fact_1697_dvd_Oantisym__conv) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4061 (all B_150 all A_178 (A_178 != zero_z126310315umeral -> (B_150 != zero_z126310315umeral -> zero_z126310315umeral != hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_178),B_150)))) # label(fact_1028_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4062 (all A_11 all N_18 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_11),N_18)),A_11) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_11),hAPP_nat_nat(suc,N_18))) # label(fact_3030_power__Suc2) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4063 (all A_92 zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_92),A_92)) # label(fact_2083_diff__self) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4064 (all N_11 all A_6 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_6)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_6),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_6),hAPP_nat_nat(suc,N_11))),one_one_real))))) # label(fact_3165_power__Suc__less__one) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4065 (all Lx_4 all Ly_2 all Rx_4 all Ry_2 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_4),Ly_2)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_4),Ry_2)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_4),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Ly_2),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_4),Ry_2)))) # label(fact_154_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4066 (all N_35 all A_182 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_182)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_35)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_182),N_35)))))) # label(fact_976_one__less__power) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4067 (all A_50 all C_15 all B_42 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,C_15),B_42)) -> hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(div_mo1740067990umeral(A_50),B_42)),C_15) = hAPP_C498520661umeral(div_mo1740067990umeral(A_50),C_15))) # label(fact_2386_mod__mod__cancel) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4068 (all A_3 all X_1 hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(ln,X_1)),hAPP_real_real(ln,A_3)) = hAPP_real_real(log(A_3),X_1)) # label(fact_3863_Log_Olog__def) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4069 (all X_11 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_11)) -> hAPP_n522471361de_int(semiri1424489471de_int,hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_11))) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_n522471361de_int(semiri1424489471de_int,hAPP_int_nat(nat,X_11))),hAPP_n522471361de_int(semiri1424489471de_int,hAPP_int_nat(nat,X_11))))) # label(fact_2814_of__nat__double) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4070 (all C_63 all D_10 all A_129 all B_108 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_129),B_108)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_63),D_10)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_129)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_63)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_129),C_63)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_108),D_10)))))))) # label(fact_1427_mult__le__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4071 (all K_1 all L (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K_1)),hAPP_int_int(bit1,L))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),L)))) # label(fact_273_rel__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4072 (all M all I all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,M),N_1)),I) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),I)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),I))) # label(fact_5120_min__diff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4073 (all A all N (zero_zero_real = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A),N) <-> zero_zero_real = A & zero_zero_nat != N)) # label(fact_548_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4074 (all N_60 all X_41 all Y_33 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_41),Y_33)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_41),N_60)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_33),N_60))))) # label(fact_589_dvd__power__same) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4075 (all N_1 (is_int(N_1) -> N_1 = quickcheck_int_of(hAPP_i1732201573de_int(quickcheck_of_int,N_1)))) # label(fact_4556_int__of__of__int) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4076 (all N_12 all R_6 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),R_6)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,R_6),one_on1645066479umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,R_6),hAPP_nat_nat(suc,N_12))),R_6))))) # label(fact_3158_realpow__Suc__le__self) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4077 (all N all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),Ma)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(real_nat,Ma))))) # label(fact_3649_real__of__nat__le__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4078 (all P_5 (hBOOL(hAPP_nat_bool(prime,P_5)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),P_5)) & (all M_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M_1),P_5)) -> one_one_nat = M_1 | M_1 = P_5)))) # label(fact_4183_prime__def) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4079 (all P all Q all R hAPP_i1948725293t_bool(hAPP_f1048215610t_bool(hAPP_f472159229t_bool(cOMBC_1683390479t_bool,P),Q),R) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool__015) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4080 (all A_28 (zero_z891286103de_int != A_28 -> one_on1684967323de_int = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_28),A_28))) # label(fact_2588_div__self) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4081 (all D collec1347809874nt_int(produc93441119t_bool(hAPP_f428220345t_bool(hAPP_f654702867t_bool(cOMBB_591320580ol_int,hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,D)))),ord_less_int))) = int_ge_less_than2(D)) # label(fact_4860_int__ge__less__than2__def) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4082 (all A_107 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_107),zero_zero_complex) = A_107) # label(fact_1967_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4083 (all N all Ma (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,N)),hAPP_int_real(real_int,Ma))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,N)),Ma)))) # label(fact_4444_number__of__le__real__of__int__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4084 (all C all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C),A)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C),B))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)))) # label(fact_2069_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4085 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),B)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B)),one_one_rat))))) # label(fact_4727_Fract__le__one__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4086 (all Y_1 all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(nat_tsub(X_1),Y_1)))))) # label(fact_2797_Nat__Transfer_Otransfer__nat__int__function__closures_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4087 (all Ma all N (Ma = N <-> hAPP_nat_complex(semiri2020571505omplex,Ma) = hAPP_nat_complex(semiri2020571505omplex,N))) # label(fact_2835_of__nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4088 (all P all Q all R hAPP_i345030060t_bool(hAPP_f725770969t_bool(hAPP_f983569339t_bool(cOMBB_550562460ol_int,P),Q),R) = hAPP_f1791153283t_bool(P,hAPP_int_fun_int_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc___018) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4089 (all A_3 all B_2 all P_3 all Q_2 (hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2) = normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_3),B_2)) -> hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,P_3),Q_2))) # label(fact_4691_normalize__eq) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4090 (all N_56 all A_206 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_206)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_206),N_56))))) # label(fact_650_zero__le__power) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4091 (all W_8 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(number_number_of_int,W_8))),hAPP_int_int(number_number_of_int,W_8)) = hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,W_8))) # label(fact_448_number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4092 (all A_64 all B_56 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_64),B_56)),B_56) = hAPP_int_int(div_mod_int(A_64),B_56)) # label(fact_2342_mod__add__self2) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4093 (all Z_2 all Y_2 all X_2 all X_3 (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),X_3)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_2),X_2)) -> ((all X (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X),X_3)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Z_2)))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_2),comple124823625p_real(X_3))))))) # label(fact_4647_SupInf_OSup__upper2) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4094 (all Ta all A_1 all D_1 (is_int(Ta) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ta),A_1)) -> (all X (is_int(X) -> ((all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A_1)) -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) != X)))))) -> (Ta != X -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),D_1) != Ta)))))))) # label(fact_5098_aset_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4095 (all N_25 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(semiri984289939at_nat,N_25)))) # label(fact_2910_of__nat__0__le__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4096 (all X_1 all Y_1 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_1)),hAPP_real_real(cos,Y_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X_1)),hAPP_real_real(sin,Y_1))) = hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),Y_1))) # label(fact_3682_cos__diff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4097 (all X_1 all Y_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> hAPP_real_real(hAPP_n546711566l_real(root,N_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),hAPP_real_real(hAPP_n546711566l_real(root,N_1),Y_1)))) # label(fact_3906_real__root__mult) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4098 (all Ma all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,N))))) # label(fact_2954_of__nat__le__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4099 (all A_35 zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,zero_zero_int),A_35)) # label(fact_2541_div__0) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4100 (all V_2 ((hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> zero_zero_int = hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(number_number_of_nat,V_2))) & (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_int_int(number_number_of_int,V_2)))) # label(fact_3284_int__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4101 (all N (hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(div_mod_nat(N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) <-> hBOOL(hAPP_nat_bool(even_odd_even_nat,N)))) # label(fact_3785_even__even__mod__4__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4102 (all A_228 A_228 = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_228),one_on1684967323de_int)) # label(fact_332_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4103 (all X_1 all Y_1 hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_1) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Y_1),hAPP_int_int(div_mod_int(X_1),Y_1))) # label(fact_4986_gcd__red__int) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4104 (all P (is_bool(P) -> fFalse = P | P = fTrue)) # label(help_If_3_1_If_000tc__RealDef__Oreal_T) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4105 (all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),Y_2)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_int_rat(number_number_of_rat,Y_2))))) # label(fact_764_le__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4106 (all P_1 image_nat_int(semiri1621563631at_int,collect_nat(hAPP_f1154658180t_bool(hAPP_f1361476881t_bool(cOMBB_int_bool_nat,P_1),semiri1621563631at_int))) = collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),P_1))) # label(fact_4733_Nat__Transfer_Otransfer__int__nat__set__functions_I5_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4107 (all A_163 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,one_one_int),A_163))) # label(fact_1153_one__dvd) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4108 (all Z_1 hAPP_complex_complex(cnj,hAPP_int_complex(ring_11397209091omplex,Z_1)) = hAPP_int_complex(ring_11397209091omplex,Z_1)) # label(fact_4578_complex__cnj__of__int) # label(axiom) # label(non_clause). [assumption]. 5.69/5.72 4109 (all B_1 all A_1 (hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1)) -> (hBOOL(hAPP_f448129468l_bool(nat_nat_set,B_1)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,image_int_nat(nat,A_1)),image_int_nat(nat,B_1))) <-> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_1),B_1)))))) # label(fact_4620_transfer__int__nat__set__relations_I5_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4110 (all V_12 all W_10 hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_12),W_10)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_12)),hAPP_int_rat(number_number_of_rat,W_10))) # label(fact_440_add__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4111 (all A_75 hAPP_C498520661umeral(div_mo1740067990umeral(A_75),A_75) = zero_z126310315umeral) # label(fact_2304_mod__self) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4112 (all A_222 A_222 = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_222),hAPP_int_complex(number528085621omplex,pls))) # label(fact_419_add__numeral__0__right) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4113 (all Diff all F all N all H (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hAPP_n546711566l_real(Diff,zero_zero_nat) = F -> ((all M_1 all T (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),T)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),H)) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),T),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T))))) -> (exists T (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),T)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,T),H)) & hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,H))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),N))) = hAPP_real_real(F,H)))))))) # label(fact_4938_Maclaurin) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4114 (all X hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,plus_plus_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),sin)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),cos)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(cos,X))),hAPP_real_real(sin,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(cos,X))),hAPP_real_real(sin,X)))))) # label(fact_4801_DERIV__sin__circle__all) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4115 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,K1)),hAPP_int_int(bit1,K2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)))) # label(fact_110_less__eq__int__code_I16_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4116 (all Ta all B_1 all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Ta),one_one_int)),B_1)) -> (all X (is_int(X) -> ((all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B_1)) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) != X)))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ta),X)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ta),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),D_1)))))))))) # label(fact_5090_bset_I8_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4117 (all N_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N_1))) # label(fact_1752_less__eq__nat_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4118 (all C_38 all A_95 all B_78 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_95),B_78)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_38),A_95)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_38),B_78))))) # label(fact_2056_add__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4119 (all N_1 all A_3 hAPP_complex_real(im,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,cis(A_3)),N_1)) = hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),A_3))) # label(fact_4143_sin__n__Im__cis__pow__n) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4120 (all C_38 all A_95 all B_78 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_95),B_78)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_38),A_95)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_38),B_78))))) # label(fact_2057_add__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4121 (all D all Ta (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),Ta)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(uminus_uminus_int,Ta))))) # label(fact_3520_uminus__dvd__conv_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4122 (all N_1 all K_2 (is_int(K_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2)) -> K_2 = hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N_1)),K_2)))) # label(fact_4510_zgcd__zmult__eq__self2) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4123 (all N_1 (hAPP_int_int(number_number_of_int,min) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),N_1) | hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),N_1) = one_one_int)) # label(fact_1625_neg__one__power) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4124 (all A (hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially)) -> (hBOOL(monoseq_real(A)) -> hBOOL(tendsto_nat_real(hAPP_f292751352t_real(hAPP_f1607377819t_real(cOMBB_1533963633al_nat,hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A))),hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,ord_at4362885an_nat(zero_zero_nat)),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),sequentially))))) # label(fact_5051_summable__Leibniz_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4125 (all A_26 all C_9 all B_23 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,C_9),B_23)) -> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_26),B_23)),C_9) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_26),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,B_23),C_9)))) # label(fact_2594_div__mult__swap) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4126 (all X_1 all A_3 all B_2 hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3)),B_2) = hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),B_2)),A_3)) # label(fact_4382_powr__powr__swap) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4127 (all Y_1 hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_real_real(tan,Y_1))) # label(fact_3618_tan__inverse) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4128 (all Lx all Rx all Ry hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx),Ry)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx),Ry))) # label(fact_181_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4129 (all X_1 X_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),zero_zero_nat)) # label(fact_4338_gcd__0__nat) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4130 (all N_1 ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N_1),zero_zero_int)) -> hAPP_int_int(zfact,N_1) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N_1),hAPP_int_int(zfact,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,N_1),one_one_int)))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N_1),zero_zero_int)) -> hAPP_int_int(zfact,N_1) = one_one_int))) # label(fact_2228_zfact_Osimps) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4131 (all C_71 all B_123 all A_146 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_123),A_146)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_71),zero_zero_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_146),C_71)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_123),C_71)))))) # label(fact_1259_mult__strict__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4132 (all Z_1 hAPP_complex_real(im,of_real_complex(Z_1)) = zero_zero_real) # label(fact_4093_Im__complex__of__real) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4133 (all M all N_1 all K_2 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(legacy_zgcd(M),N_1)),hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),N_1)))) # label(fact_4507_zgcd__zdvd__zgcd__zmult) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4134 (all P all Q all R hAPP_real_bool(P,hAPP_int_real(Q,R)) = hAPP_int_bool(hAPP_f429384717t_bool(hAPP_f1173902867t_bool(cOMBB_real_bool_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__HOL__Obool_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4135 (all K_2 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,min),K_2) = hAPP_int_int(pred,K_2)) # label(fact_4047_add__Min) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4136 (all Y_2 all X_2 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),X_2)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),Y_2)) -> (zero_z126310315umeral = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X_2),Y_2) <-> zero_z126310315umeral = X_2 & zero_z126310315umeral = Y_2)))) # label(fact_2166_add__nonneg__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4137 (all Ma all N ((hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) | zero_zero_nat = N -> divmod_nat(Ma,N) = hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,zero_zero_nat),Ma)) & (-(N = zero_zero_nat | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N))) -> hAPP_P1586233937at_nat(produc1391996073at_nat(hAPP_f837737749at_nat(hAPP_f999082675at_nat(cOMBB_348438404at_nat,product_Pair_nat_nat),suc)),divmod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),N),N)) = divmod_nat(Ma,N)))) # label(fact_4850_divmod__nat__if) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4138 (all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),pls)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,K_1)),pls)))) # label(fact_125_rel__simps_I29_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4139 (all X_2 all Y_2 (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_2),Y_2)) -> (X_2 != Y_2 -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,X_2),Y_2))))) # label(fact_1015_order__le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4140 (all B all A (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A),B)))))) # label(fact_2631_nonneg1__imp__zdiv__pos__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4141 (all A_1 hAPP_f957591787ol_nat(finite_card_int,image_nat_int(semiri1621563631at_int,A_1)) = hAPP_f22106695ol_nat(finite_card_nat,A_1)) # label(fact_4604_Nat__Transfer_Otransfer__nat__int__set__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4142 (all A_186 all K_8 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_186),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),K_8))),zero_zero_rat)) -> A_186 = zero_zero_rat)) # label(fact_901_even__power__le__0__imp__0) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4143 (all A_3 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),hAPP_int_int(div_mod_int(A_3),B_2))))) # label(fact_2436_neg__mod__bound) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4144 (all W_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,W_1)),zero_zero_int)))) # label(fact_712_bin__less__0__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4145 (all X_1 (is_int(X_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),zero_zero_int)) -> (exists N_2 hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_2))) = X_1)))) # label(fact_3556_negD) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4146 (all Z_1 (hBOOL(hAPP_int_bool(nat_neg,Z_1)) -> hAPP_int_nat(nat,Z_1) = zero_zero_nat)) # label(fact_3278_neg__nat) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4147 (all C_88 all A_180 all B_152 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_180),B_152)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,B_152),C_88)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_180),C_88))))) # label(fact_985_dvd__trans) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4148 (all M hAPP_nat_nat(suc,M) != zero_zero_nat) # label(fact_2851_Zero__neq__Suc) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4149 (all X_37 all Y_29 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_29),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_37),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),X_37)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_29)) -> X_37 = Y_29)))) # label(fact_790_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4150 (all Y_2 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_n546711566l_real(root,N),Y_2)))))) # label(fact_3917_real__root__ge__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4151 (all M all N_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,M),N_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(fact_fact_int,M)),hAPP_int_int(fact_fact_int,N_1))))) # label(fact_4271_fact__mono__int) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4152 (all X_23 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_23),X_23)),one_one_real) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_23),one_one_real)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_23),one_one_real))) # label(fact_1544_real__squared__diff__one__factored) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4153 (all A_89 A_89 = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,one_on1645066479umeral),A_89)) # label(fact_2117_mult__1) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4154 (all A_3 all B_2 all N_1 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(legacy_zgcd(A_3),B_2)),N_1) = hAPP_int_int(legacy_zgcd(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N_1)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),N_1))) # label(fact_4493_zgcd__power__distrib) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4155 (all K_2 hAPP_int_int(bit1,K_2) != pls) # label(fact_348_rel__simps_I46_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4156 (all X_1 -(hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),X_1)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),X_1)))) # label(fact_1690_dvd_Oless__irrefl) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4157 (all A_3 all B_2 ((zero_zero_int != hAPP_int_int(div_mod_int(A_3),B_2) -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(div_mod_int(A_3),B_2)),B_2) = hAPP_int_int(div_mod_int(A_3),hAPP_int_int(uminus_uminus_int,B_2))) & (zero_zero_int = hAPP_int_int(div_mod_int(A_3),B_2) -> zero_zero_int = hAPP_int_int(div_mod_int(A_3),hAPP_int_int(uminus_uminus_int,B_2))))) # label(fact_3543_zmod__zminus2__eq__if) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4158 (all A_236 all B_165 all C_96 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_236),B_165)),C_96) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_236),C_96)),B_165)) # label(fact_217_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4159 (all A_177 zero_zero_complex = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_177),zero_zero_complex)) # label(fact_1040_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4160 (all A (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),A)),zero_zero_rat)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A),zero_zero_rat)))) # label(fact_689_even__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4161 (all K_2 all M all N_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(div_mod_int(M),N_1))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),N_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M))))) # label(fact_2403_zdvd__zmod__imp__zdvd) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4162 (all X_2 all Y_2 all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B),X_2)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B),Y_2))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),Y_2))))) # label(fact_534_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4163 (all A all B (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A),zero_zero_rat)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B)) | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B),zero_zero_rat)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A)))) # label(fact_1247_mult__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4164 (all M_11 -hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(semiri1619134803umeral,M_11)),zero_z126310315umeral))) # label(fact_2920_of__nat__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4165 (all I all K_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),K_2)) -> (hAPP_nat_nat(suc,I) != K_2 -> -(all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_1)) -> hAPP_nat_nat(suc,J_1) != K_2))))) # label(fact_3292_lessE) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4166 (all A_127 all C_61 all B_106 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_127),C_61)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_106),C_61))) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_61)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_127),B_106))))) # label(fact_1442_mult__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4167 (all V_7 all B_99 all C_55 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_7)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_99),C_55)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_7)),B_99)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_7)),C_55))) # label(fact_1556_right__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4168 (all P all Q all R hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(hAPP_f465821005nt_int(cOMBC_1964283556nt_int,P),Q),R) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Int__Oint_000tc__prod_Itc__Int__Oin) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4169 (all A all C all B (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),C))) <-> zero_zero_rat = C | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A),B)))) # label(fact_1380_dvd__mult__cancel__right) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4170 (all A_75 zero_zero_int = hAPP_int_int(div_mod_int(A_75),A_75)) # label(fact_2302_mod__self) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4171 (all A_3 all B_2 all Q_2 all R_3 (is_int(Q_2) & is_int(B_2) -> (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3))) -> (zero_zero_int != B_2 -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) = Q_2)))) # label(fact_2794_divmod__int__rel__div) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4172 (all N_16 all M_5 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_16),M_5)) -> hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_nat_real(semiri132038758t_real,M_5)),hAPP_nat_real(semiri132038758t_real,N_16)) = hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_5),N_16)))) # label(fact_3103_of__nat__diff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4173 (all X_2 all P_5 (is_int(X_2) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),hAPP_i1948725293t_bool(sRStar,P_5))) -> X_2 = hAPP_int_int(standardRes(P_5),X_2)))) # label(fact_2511_StandardRes__SRStar__prop3) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4174 (all M all N_1 all R_5 all S_3 all S_1 all T_2 all T_1 all Rn all Sn_1 all Tn_1 all R_3 (is_int(Rn) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),R_3)) -> (xzgcda(M,N_1,R_5,R_3,S_3,S_1,T_2,T_1) = hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,Rn),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Sn_1),Tn_1)) -> (R_5 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,S_3),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,T_2),N_1)) -> (R_3 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,S_1),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,T_1),N_1)) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Sn_1),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Tn_1),N_1)) = Rn)))))) # label(fact_4201_xzgcda__linear) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4175 (all A_3 all B_2 zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(div_mod_int(A_3),B_2)),B_2)) # label(fact_2562_zmod__zdiv__trivial) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4176 (all Q_1 all P_1 ((all A_2 all B_4 (is_int(B_4) & is_int(A_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_4)) -> (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_2),B_4) -> hBOOL(hAPP_rat_bool(P_1,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_4))))))) -> hBOOL(hAPP_rat_bool(P_1,Q_1)))) # label(fact_5026_Rat__induct) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4177 (all Y_2 all X_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_int_nat(nat,X_2)),hAPP_int_nat(nat,Y_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_2),Y_2)))))) # label(fact_2758_transfer__nat__int__relations_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4178 (all N_19 zero_z891286103de_int = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,zero_z891286103de_int),hAPP_nat_nat(suc,N_19))) # label(fact_2994_power__0__Suc) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4179 (all A_86 all B_75 (is_int(A_86) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_86),B_75)),B_75) = A_86)) # label(fact_2137_diff__add__cancel) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4180 (all B_2 all Bq all Br all A_3 all C_1 all Aq all Ar (is_int(C_1) -> (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,C_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Aq),Ar))) -> (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(B_2,C_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Bq),Br))) -> (zero_zero_int != C_1 -> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2),C_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aq),Bq)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ar),Br)),C_1))),hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ar),Br)),C_1))))))))) # label(fact_2801_zadd1__lemma) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4181 (all A_233 all C_93 all D_18 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_233),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_93),D_18)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_93),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_233),D_18))) # label(fact_239_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4182 (all M_22 all A_214 all N_62 all B_159 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_214),N_62)),B_159)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_22),N_62)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_214),M_22)),B_159))))) # label(fact_569_power__le__dvd) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4183 (all X_44 all Y_36 all Z_15 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_44),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y_36),Z_15)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_44),Y_36)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_44),Z_15))) # label(fact_322_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4184 (all L all K_1 (is_int(L) & is_int(K_1) -> (K_1 = zero_zero_int -> divmod_int(K_1,L) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),zero_zero_int)) & (zero_zero_int != K_1 -> (L = zero_zero_int -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),K_1) = divmod_int(K_1,L)) & (zero_zero_int != L -> divmod_int(K_1,L) = hAPP_P1975530577nt_int(produc713050258nt_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(sgn_sgn_int,L))),hAPP_P1975530577nt_int(hAPP_P1145851913nt_int(hAPP_b40753821nt_int(if_Pro1731782967nt_int,hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,hAPP_int_int(sgn_sgn_int,K_1)),hAPP_int_int(sgn_sgn_int,L))),pdivmod(K_1,L)),hAPP_P1975530577nt_int(produc1518849193nt_int(hAPP_f465821005nt_int(hAPP_f1081447804nt_int(cOMBS_1470252563nt_int,hAPP_f617310012nt_int(hAPP_f197974549nt_int(cOMBB_638531587nt_int,cOMBS_1891347093nt_int),hAPP_f1148171384nt_int(hAPP_f334321603nt_int(cOMBB_1484610687nt_int,hAPP_f1582586579nt_int(cOMBC_707974837nt_int,hAPP_f251264923nt_int(hAPP_f279307923nt_int(cOMBB_662120866nt_int,if_Pro1731782967nt_int),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),zero_zero_int)))),hAPP_i1584592887nt_int(hAPP_f465821005nt_int(cOMBC_1964283556nt_int,hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),uminus_uminus_int)),zero_zero_int)))),hAPP_f1574055885nt_int(hAPP_f960630521nt_int(cOMBC_1805044090nt_int,hAPP_f142232355nt_int(hAPP_f1377453843nt_int(cOMBB_320287386nt_int,cOMBB_1846624359nt_int),hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,minus_minus_int),uminus_uminus_int)),one_one_int)))),hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(abs_abs_int,L))))),pdivmod(K_1,L)))))))) # label(fact_4855_divmod__int__code) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4185 (all F all N zero_zero_real = hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),F)),hAPP_r474017924t_real(power_power_real,zero_zero_real))),hAPP_n1699378549t_bool(ord_at4362885an_nat(hAPP_nat_nat(suc,zero_zero_nat)),N))) # label(fact_4903_sumr__one__lb__realpow__zero) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4186 (all X_2 all N all Y_2 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_2),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_2),hAPP_nat_nat(suc,N)) <-> Y_2 = X_2)) # label(fact_2882_exp__mono__eq) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4187 (all P all Q all R hAPP_f54304608l_bool(hAPP_n215258509l_bool(P,R),Q) = hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4188 (all A_112 all B_91 all C_51 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_112),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_91),C_51)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_112),B_91)),C_51)) # label(fact_1904_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4189 (all A all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B)) <-> hAPP_nat_nat(div_mod_nat(B),A) = zero_zero_nat)) # label(fact_2434_dvd__eq__mod__eq__0) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4190 (all A_234 all C_94 all D_19 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_234),C_94)),D_19) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_234),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_94),D_19))) # label(fact_232_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4191 (all P_5 hAPP_P1975530577nt_int(produc1518849193nt_int(hAPP_f465821005nt_int(hAPP_f1081447804nt_int(cOMBS_1470252563nt_int,hAPP_f400200275nt_int(hAPP_f1731308757nt_int(cOMBB_735549356nt_int,cOMBB_1389251822nt_int),hAPP_P621635040nt_int(hAPP_f1582586579nt_int(cOMBC_707974837nt_int,hAPP_f251264923nt_int(hAPP_f279307923nt_int(cOMBB_662120866nt_int,if_Pro1731782967nt_int),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),zero_zero_int))),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int)))),hAPP_f1574055885nt_int(hAPP_f434654626nt_int(cOMBS_2011279539nt_int,hAPP_f2094497995nt_int(hAPP_f534361363nt_int(cOMBB_1657320178nt_int,cOMBC_1964283556nt_int),hAPP_f2004554147nt_int(hAPP_f1228585167nt_int(cOMBB_1102189010nt_int,hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int)),hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,times_times_int),sgn_sgn_int)))),abs_abs_int))),quotient_of(P_5)) = quotient_of(hAPP_rat_rat(inverse_inverse_rat,P_5))) # label(fact_4838_rat__inverse__code) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4192 (all A_239 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_239),A_239) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_239),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_40_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4193 (all X_35 all Y_27 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_35),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_27),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_846_sum__power2__ge__zero) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4194 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),zero_zero_real)) -> zero_zero_nat = natceiling(X_1))) # label(fact_3398_natceiling__neg) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4195 (all P (-hBOOL(P) | -hBOOL(hAPP_bool_bool(fNot,P)))) # label(help_fNot_1_1_U) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4196 (all A_3 all B_2 hAPP_complex_complex(invers1449016382omplex,hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),B_2)) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,A_3),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(uminus_uminus_real,B_2)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_3966_complex__inverse) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4197 (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),one_one_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sqrt,X_2)),one_one_real)))) # label(fact_3331_real__sqrt__le__1__iff) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4198 (all N_1 (hAPP_nat_nat(div_mod_nat(N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))) -> -hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),one_one_nat)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_3790_lemma__odd__mod__4__div__2) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4199 (all B_2 zero_zero_int = hAPP_int_int(div_mod_int(zero_zero_int),B_2)) # label(fact_2395_zmod__zero) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4200 (all Ta all A_1 all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ta),one_one_int)),A_1)) -> (all X (is_int(X) -> ((all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A_1)) -> X != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa))))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X),Ta)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),D_1)),Ta))))))))) # label(fact_5089_aset_I6_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4201 (all N_11 all A_6 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_6)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_6),one_on1684967323de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_6),hAPP_nat_nat(suc,N_11))),one_on1684967323de_int))))) # label(fact_3163_power__Suc__less__one) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4202 (all A_3 all B_2 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2) != zero_zero_nat -> (exists A_4 exists B_3 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_4),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)) = A_3 & B_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)) & one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_4),B_3))))) # label(fact_4372_gcd__coprime__exists__nat) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4203 (all M_27 all A_224 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_224),one_one_nat)),M_27) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_27),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_224),M_27))) # label(fact_400_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4204 (all X_3 (hBOOL(summable_complex(X_3)) -> hBOOL(summable_complex(hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,cnj),X_3))))) # label(fact_4844_cnj_Osummable) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4205 (all A_88 A_88 = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,one_one_complex),A_88)) # label(fact_2125_mult__1__left) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4206 (all X_1 all Y_1 all L_2 all M hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_1)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(uminus_uminus_real,L_2)),hAPP_real_real(uminus_uminus_real,M))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_real_real(uminus_uminus_real,L_2)))),hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Y_1),hAPP_real_real(uminus_uminus_real,M))))))) # label(fact_3495_abs__sum__triangle__ineq) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4207 (all M all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N_1) = hAPP_P1654798560at_nat(product_fst_nat_nat,divmod_nat(M,N_1))) # label(fact_4238_div__nat__def) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4208 (all U_1 all X_1 (X_1 = hAPP_real_real(exp_real,U_1) -> U_1 = hAPP_real_real(ln,X_1))) # label(fact_3820_exp__ln__eq) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4209 (all N (zero_zero_nat = N <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(suc,zero_zero_nat))))) # label(fact_3049_less__Suc0) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4210 (all X_1 all Y_1 (is_int(Y_1) & is_int(X_1) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),X_1)) | X_1 = Y_1 | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1)))) # label(fact_66_zless__linear) # label(axiom) # label(non_clause). [assumption]. 5.69/5.73 4211 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(div_mod_nat(M),N_1)),N_1)))) # label(fact_2512_mod__less__divisor) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4212 (all Ma all N (is_int(Ma) & is_int(N) -> (zero_zero_int = N & Ma = zero_zero_int <-> zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Ma),N)))) # label(fact_4977_gcd__zero__int) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4213 (all M_26 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat)),M_26) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_26),M_26)) # label(fact_407_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4214 (all M M = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),hAPP_nat_nat(suc,zero_zero_nat))) # label(fact_3079_div__1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4215 (all M all X_1 (is_int(X_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),M)) -> zero_zero_int = X_1))))) # label(fact_1488_Int2_Ozcong__zero) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4216 (all R_3 all P_3 all Q_2 (normalize(R_3) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2) -> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,P_3),Q_2) = one_one_int)) # label(fact_5017_normalize__coprime) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4217 (all M hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),M))))) # label(fact_1788_le__cube) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4218 (all B_2 all A_3 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_2),A_3))),hAPP_complex_real(norm_norm_complex,B_2))),hAPP_complex_real(norm_norm_complex,A_3)))) # label(fact_3368_complex__mod__triangle__ineq2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4219 (all I all J_2 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(legacy_zgcd(I),J_2)),J_2))) # label(fact_4483_zgcd__zdvd2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4220 (all Ma all N all K_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1)) -> (Ma = N <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N)))) # label(fact_1844_nat__mult__eq__cancel1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4221 (all X_1 all Y_1 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) -> Y_1 != X_1)) # label(fact_1712_dvd_Oless__imp__not__eq2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4222 (all X_2 all N (hBOOL(hAPP_int_bool(even_odd_even_int,X_2)) & N != zero_zero_nat <-> hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_2),N))))) # label(fact_3092_even__power) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4223 (all A_202 all N_52 all N_51 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_52),N_51)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_202)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_202),one_one_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_202),N_51)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_202),N_52))))))) # label(fact_676_power__decreasing) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4224 (all C_89 all D_17 all A_200 all B_154 all R_8 (R_8 != zero_zero_complex -> (A_200 = B_154 & C_89 != D_17 -> hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_154),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,R_8),D_17)) != hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_200),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,R_8),C_89))))) # label(fact_696_add__scale__eq__noteq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4225 (all N hAPP_nat_real(cos_coeff,N) = hAPP_nat_real(diffs_real(sin_coeff),N)) # label(fact_4873_sin__fdiffs2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4226 (all P_5 hAPP_P1975530577nt_int(produc1518849193nt_int(hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),abs_abs_int)),quotient_of(P_5)) = quotient_of(hAPP_rat_rat(abs_abs_rat,P_5))) # label(fact_4835_rat__abs__code) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4227 (all N all Ma (is_int(Ma) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),N)) <-> hAPP_int_int(legacy_zgcd(N),Ma) = Ma)))) # label(fact_4511_zdvd__iff__zgcd) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4228 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(cos,X_1)))))) # label(fact_3589_cos__ge__zero) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4229 (all V_3 all V_2 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls)) -> zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3))) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls)) -> hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_2),V_3)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3))))) # label(fact_1009_mult__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4230 (all K_2 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),one_one_int) = hAPP_int_int(succ,K_2)) # label(fact_3298_succ__def) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4231 (all B_117 all A_140 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_117),A_140))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_140)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_117))))) # label(fact_1292_zero__less__mult__pos2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4232 (all M_17 all N_30 ((N_30 = zero_zero_nat -> one_one_complex = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,M_17),N_30)) & (N_30 != zero_zero_nat -> hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,M_17),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,M_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_30),one_one_nat))) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,M_17),N_30)))) # label(fact_1833_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4233 (all A_104 A_104 = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,zero_z126310315umeral),A_104)) # label(fact_1987_add__0__left) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4234 (all M all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,M)))),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),M)),N_1)) # label(fact_3207_Suc__div__eq__add3__div) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4235 (all P_1 all K_1 (hBOOL(hAPP_nat_bool(P_1,K_1)) -> ((all N_2 (hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,N_2))) -> hBOOL(hAPP_nat_bool(P_1,N_2)))) -> hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))))) # label(fact_3310_zero__induct) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4236 (all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_nat_bool(even_odd_even_nat,N)) <-> -hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)))))) # label(fact_3778_even__num__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4237 (all X_1 all N_5 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_5)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),hAPP_real_real(hAPP_n546711566l_real(root,N_5),X_1)))))))) # label(fact_3937_real__root__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4238 (all Nat hAPP_nat_nat(nat_size,hAPP_nat_nat(suc,Nat)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(nat_size,Nat)),hAPP_nat_nat(suc,zero_zero_nat))) # label(fact_4055_nat_Osize_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4239 (all X_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,X_2)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),pls)))) # label(fact_759_less__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4240 (all N all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,N)),Ma)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,N)),hAPP_int_real(real_int,Ma))))) # label(fact_4442_number__of__less__real__of__int__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4241 (all A_183 one_one_real = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_183),zero_zero_nat)) # label(fact_957_power__0) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4242 (all M zero_zero_nat != hAPP_nat_nat(suc,M)) # label(fact_2850_Suc__neq__Zero) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4243 (all M all Y_1 all X_1 (is_int(Y_1) & is_int(X_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Y_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),Y_1),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),M)) -> X_1 = Y_1)))))))) # label(fact_1394_zcong__less__eq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4244 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) -> -hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(d22set,A))))) # label(fact_2270_d22set__le__swap) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4245 (all A_234 all C_94 all D_19 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_234),C_94)),D_19) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_234),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_94),D_19))) # label(fact_227_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4246 (all N_59 all A_212 (A_212 != zero_zero_complex -> hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_212),N_59) != zero_zero_complex)) # label(fact_592_field__power__not__zero) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4247 (all A all A_1 all Ma (hBOOL(hAPP_int_bool(is_RRset(A_1),Ma)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),A_1)) -> one_one_int = hAPP_int_int(legacy_zgcd(A),Ma)))) # label(fact_4588_RRset__gcd) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4248 (all X_1 hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_1),hAPP_real_real(abs_abs_real,X_1)) = hAPP_real_real(sgn_sgn_real,X_1)) # label(fact_3564_real__sgn__eq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4249 (all X_21 all N_31 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31)) -> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_21),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_31),one_one_nat))),X_21) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_21),N_31))) # label(fact_1734_realpow__minus__mult) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4250 (all X_1 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(arctan,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))))) = hAPP_real_real(arctan,X_1)) # label(fact_3372_arctan__half) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4251 (all X_1 all B_2 all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> (one_one_real != A_3 -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_2)) -> (one_one_real != B_2 -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> hAPP_real_real(log(A_3),X_1) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(ln,B_2)),hAPP_real_real(ln,A_3))),hAPP_real_real(log(B_2),X_1)))))))) # label(fact_3875_log__eq__div__ln__mult__log) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4252 (all W_17 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_i1732201573de_int(number1226105091de_int,W_17)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_i1732201573de_int(number1226105091de_int,W_17)),hAPP_i1732201573de_int(number1226105091de_int,W_17))) # label(fact_15_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4253 (all K_1 all Ma all N (hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),Ma) <-> Ma = N | K_1 = zero_zero_nat)) # label(fact_1756_mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4254 (all X_1 all N_1 hAPP_real_real(tan,X_1) = hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),pi)))) # label(fact_3690_tan__periodic__nat) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4255 (all A_227 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,one_one_rat),A_227) = A_227) # label(fact_340_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4256 (all R_3 all X_1 hAPP_complex_complex(scaleR1652505878omplex(R_3),X_1) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_complex_real(re,X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_complex_real(im,X_1)))) # label(fact_5114_complex__scaleR__def) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4257 (all A_238 all B_167 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_238),B_167) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_167),A_238)) # label(fact_188_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4258 (all N all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Ma)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),Ma)),Ma)) <-> N = one_one_nat))) # label(fact_1631_dvd__mult__cancel2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4259 (all B_126 all A_150 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_150)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_126),zero_z891286103de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_150),B_126)),zero_z891286103de_int))))) # label(fact_1237_mult__nonneg__nonpos) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4260 (all M (hAPP_nat_nat(div_mod_nat(M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = one_one_nat | hAPP_nat_nat(div_mod_nat(M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))) | hAPP_nat_nat(div_mod_nat(M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) | zero_zero_nat = hAPP_nat_nat(div_mod_nat(M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_2528_mod__exhaust__less__4) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4261 (all N_35 all A_182 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_182)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_35)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_182),N_35)))))) # label(fact_974_one__less__power) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4262 (all C_65 all D_12 all A_131 all B_110 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_131),B_110)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_65),D_12)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_131)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_65)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_131),C_65)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_110),D_12)))))))) # label(fact_1413_mult__strict__mono_H) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4263 (all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),zero_zero_int)) -> zero_zero_nat = hAPP_int_nat(nat,Z_1))) # label(fact_2738_nat__le__0) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4264 (all N_1 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N_1)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,X_1)),one_one_real))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N_1),X_1)))) # label(fact_4446_lemma__floor2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4265 (all R_3 hAPP_real_complex(hAPP_r265291036omplex(complex,zero_zero_real),R_3) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),of_real_complex(R_3))) # label(fact_3995_i__complex__of__real) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4266 (all A_3 (hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = one_one_int -> hAPP_int_int(abs_abs_int,A_3) = one_one_int)) # label(fact_3462_power2__eq1__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4267 (all Y_6 all P_6 all N_10 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_6),N_10)) -> hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N_10)),P_6)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),P_6))),Y_6))) # label(fact_3175_lemma__realpow__diff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4268 (all B_94 all C_52 all A_115 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_115)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_94),C_52)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_94),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_115),C_52)))))) # label(fact_1879_add__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4269 (all B_97 all A_118 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_118),zero_z126310315umeral)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_97),zero_z126310315umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_118),B_97)),zero_z126310315umeral))))) # label(fact_1856_add__nonpos__neg) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4270 (all X_1 hAPP_complex_real(im,hAPP_complex_complex(cnj,X_1)) = hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,X_1))) # label(fact_4092_complex__Im__cnj) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4271 (all N_1 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N_1),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N_1),M)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,M),zero_zero_int)))) # label(fact_1395_zdvd__bounds) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4272 (all Ma all N (one_one_nat = N & one_one_nat = Ma <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),N) = one_one_nat)) # label(fact_1794_nat__1__eq__mult__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4273 (all X_2 hAPP_real_real(ln,X_2) = the_real(hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),exp_real)),X_2))) # label(fact_4883_ln__def) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4274 (all Lx_1 all Rx_1 all Ry_1 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_1),Rx_1)),Ry_1) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_1),Ry_1))) # label(fact_174_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4275 (all A_233 all C_93 all D_18 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_233),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_93),D_18)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_93),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_233),D_18))) # label(fact_233_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4276 (all N all A (hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially)) -> ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(A,N_2)))) -> ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A,hAPP_nat_nat(suc,N_2))),hAPP_nat_real(A,N_2)))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A))),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N)),one_one_nat))))))))) # label(fact_5053_summable__Leibniz_H_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4277 (all Z_1 all W ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W))) -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,W))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),hAPP_int_nat(number_number_of_nat,W)))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),hAPP_int_nat(number_number_of_nat,W)))) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W))) -> one_one_int = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,W)))))) # label(fact_4788_zpower__number__of__odd) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4278 (all Ma all X_2 hAPP_i1948725293t_bool(noXRRset(Ma),X_2) = image_int_int(hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,times_times_int),X_2),hAPP_i1948725293t_bool(norRRset,Ma))) # label(fact_4771_noXRRset__def) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4279 (all M_27 all A_224 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,M_27),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_224),M_27)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_224),one_one_complex)),M_27)) # label(fact_403_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4280 (all N_1 all X_1 (X_1 = hAPP_int_real(real_int,archim1246769320r_real(X_1)) -> archim1246769320r_real(hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,archim1246769320r_real(X_1)),N_1))) # label(fact_4424_floor__power) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4281 (all P all Q all R hAPP_r1392521531t_bool(hAPP_f332806115t_bool(hAPP_f1177545951t_bool(cOMBB_1459832681l_real,P),Q),R) = hAPP_f1576199059t_bool(P,hAPP_r847664445t_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__RealDef__Oreal_J_Mt) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4282 (all C_41 all A_98 all B_81 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_41),A_98)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_41),B_81))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_98),B_81)))) # label(fact_2037_add__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4283 (all A_238 all B_167 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_167),A_238) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_238),B_167)) # label(fact_191_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4284 (all K_2 all L_2 hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),L_2)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit0,K_2)),hAPP_int_int(bit1,L_2))) # label(fact_446_add__Bit0__Bit1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4285 (all K_6 all M_16 all N_29 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_6),hAPP_nat_nat(div_mod_nat(M_16),N_29))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_6),N_29)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_6),M_16))))) # label(fact_2381_dvd__mod__imp__dvd) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4286 (all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),Z_1))))) # label(fact_768_le__imp__0__less) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4287 (all B_90 all A_111 all C_50 (hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_90),A_111) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_50),A_111) -> C_50 = B_90)) # label(fact_1915_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4288 (all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),one_one_nat) = N_1) # label(fact_1795_nat__mult__1__right) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4289 (all W_1 all X_2 (is_int(X_2) -> (X_2 = hAPP_int_int(number_number_of_int,W_1) <-> hAPP_int_int(number_number_of_int,W_1) = X_2))) # label(fact_252_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4290 (all A_110 all B_89 all C_49 (is_int(B_89) & is_int(C_49) -> (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_110),B_89) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_110),C_49) -> B_89 = C_49))) # label(fact_1918_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4291 (all J_2 all I -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),I)),I))) # label(fact_1765_not__add__less2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4292 (all K_2 all M all N_1 hAPP_int_int(legacy_zgcd(K_2),hAPP_int_int(legacy_zgcd(M),N_1)) = hAPP_int_int(legacy_zgcd(M),hAPP_int_int(legacy_zgcd(K_2),N_1))) # label(fact_4491_zgcd__left__commute) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4293 (all X_37 all Y_29 (hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_29),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_37),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),X_37)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),Y_29)) -> Y_29 = X_37)))) # label(fact_792_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4294 (all C all A all B (is_int(C) -> (zero_zero_int = C | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),B)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),A)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B)))))) # label(fact_1377_dvd__mult__cancel__left) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4295 (all B_55 all A_63 hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_55),A_63)),B_55) = hAPP_nat_nat(div_mod_nat(A_63),B_55)) # label(fact_2347_mod__add__self1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4296 (all A_128 all C_62 all B_107 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_128),C_62)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_107),C_62))) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_62)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_128),B_107))))) # label(fact_1436_mult__right__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4297 (all A_35 zero_z126310315umeral = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,zero_z126310315umeral),A_35)) # label(fact_2543_div__0) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4298 (all Z_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Z_1),Z_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_1)) # label(fact_105_nat__mult__2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4299 (all C_8 all A_24 all B_21 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_24),B_21)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_21),C_8)),A_24) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B_21),A_24)),C_8))) # label(fact_2604_dvd__div__mult) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4300 (all X_1 all Y_1 hAPP_P1975530577nt_int(hAPP_P1145851913nt_int(hAPP_b40753821nt_int(if_Pro1731782967nt_int,fFalse),X_1),Y_1) = Y_1) # label(help_If_2_1_If_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_T) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4301 (all C_70 all A_145 all B_122 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_145),B_122)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_70)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_70),A_145)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_70),B_122)))))) # label(fact_1263_comm__mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4302 (all P_3 normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),zero_zero_int)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int)) # label(fact_4660_normalize__denom__zero) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4303 (all N_35 all A_182 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_182)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_35)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_182),N_35)))))) # label(fact_973_one__less__power) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4304 (all P all Q all R hAPP_r1043989325l_real(hAPP_f1161935955l_real(hAPP_f328572565l_real(cOMBB_520243667l_real,P),Q),R) = hAPP_f183342685l_real(P,hAPP_r794344869l_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__fun_Itc__prod_Itc__Real_045) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4305 (all A_53 all B_45 all C_17 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(div_mod_int(A_53),C_17)),hAPP_int_int(div_mod_int(B_45),C_17))),C_17) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_53),B_45)),C_17)) # label(fact_2376_mod__diff__eq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4306 (all M all N_1 (is_int(N_1) -> (zero_zero_int != N_1 -> hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M),N_1)),N_1) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,M),N_1)),one_one_rat)))) # label(fact_4722_Fract__add__one) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4307 (all X_2 all N all Diff all F (hAPP_n546711566l_real(Diff,zero_zero_nat) = F -> ((all M_1 all T (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_2))) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N)) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),T),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T))))) -> (exists T (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_2))) & hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,X_2))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),N))) = hAPP_real_real(F,X_2)))))) # label(fact_4932_Maclaurin__bi__le) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4308 (all Z_1 all X_1 all Y_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),Z_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Z_1))))) # label(fact_1704_dvd_Oorder__trans) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4309 (all W_16 hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit1,pls)),W_16)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,one_one_complex),hAPP_int_complex(number528085621omplex,W_16))) # label(fact_52_add__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4310 (all P_1 all L ((exists X (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,L),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X),zero_zero_complex))) & hBOOL(hAPP_complex_bool(P_1,X)))) <-> (exists X hBOOL(hAPP_complex_bool(P_1,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,L),X)))))) # label(fact_1808_unity__coeff__ex) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4311 (all A_9 all N_15 all B_8 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_9),hAPP_nat_nat(suc,N_15))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_8),hAPP_nat_nat(suc,N_15)))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_8)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_9),B_8))))) # label(fact_3109_power__le__imp__le__base) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4312 (all A_3 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),A_3)),one_one_int),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),one_one_int),P_3)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),P_3)))))) # label(fact_1609_zcong__square) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4313 (all B_58 all B_57 all A_66 all C_26 all A_65 (hAPP_C498520661umeral(div_mo1740067990umeral(A_66),C_26) = hAPP_C498520661umeral(div_mo1740067990umeral(A_65),C_26) -> (hAPP_C498520661umeral(div_mo1740067990umeral(B_58),C_26) = hAPP_C498520661umeral(div_mo1740067990umeral(B_57),C_26) -> hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_65),B_57)),C_26) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_66),B_58)),C_26)))) # label(fact_2340_mod__mult__cong) # label(axiom) # label(non_clause). [assumption]. 5.74/5.73 4314 (all B_117 all A_140 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_117),A_140))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_140)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_117))))) # label(fact_1294_zero__less__mult__pos2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4315 (all X_36 all Y_28 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_36),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_28),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_28)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_36),Y_28))))) # label(fact_840_power2__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4316 (all B_40 all A_48 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_40),A_48)),B_40) = zero_zero_int) # label(fact_2411_mod__mult__self1__is__0) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4317 (all K_2 all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),K_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),K_2))))) # label(fact_1725_le__trans) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4318 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),zero_zero_int)) -> hAPP_int_int(div_mod_int(A_3),B_2) = hAPP_P1175774780nt_int(product_snd_int_int,hAPP_P1975530577nt_int(negateSnd,posDivAlg(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2))))))) # label(fact_4237_mod__neg__neg) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4319 (all F (hBOOL(summable_real(F)) -> ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(F,N_2)))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_f352196356l_real(suminf_real,F)))))) # label(fact_4858_suminf__ge__zero) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4320 (all M_7 all N_21 hAPP_n522471361de_int(semiri1424489471de_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_7),N_21)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(semiri1424489471de_int,M_7)),hAPP_n522471361de_int(semiri1424489471de_int,N_21))) # label(fact_2964_of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4321 (all A_236 all B_165 all C_96 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_236),C_96)),B_165) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_236),B_165)),C_96)) # label(fact_216_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4322 (all D all Sa exists Z ((all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Sa))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Sa)))))) & is_int(Z))) # label(fact_2249_pinf_I9_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4323 (all M_8 all N_22 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_nat_complex(semiri2020571505omplex,M_8)),hAPP_nat_complex(semiri2020571505omplex,N_22)) = hAPP_nat_complex(semiri2020571505omplex,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_8),N_22))) # label(fact_2958_of__nat__add) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4324 (all W_8 hAPP_int_complex(number528085621omplex,hAPP_int_int(bit1,W_8)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,one_one_complex),hAPP_int_complex(number528085621omplex,W_8))),hAPP_int_complex(number528085621omplex,W_8))) # label(fact_450_number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4325 (all Code_numeral_1 hAPP_C2129356693al_nat(size_s945831648umeral,code_S1047413653umeral(Code_numeral_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_C2129356693al_nat(size_s945831648umeral,Code_numeral_1)),hAPP_nat_nat(suc,zero_zero_nat))) # label(fact_4088_code__numeral_Osize_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4326 (all X_46 X_46 = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_46),one_one_nat)) # label(fact_78_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4327 (all B_93 all A_114 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_114)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_93)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_114),B_93)))))) # label(fact_1882_add__nonneg__pos) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4328 (all A_183 one_one_int = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_183),zero_zero_nat)) # label(fact_955_power__0) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4329 (all X ((-hBOOL(hAPP_nat_bool(even_odd_even_nat,X)) -> hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,X),hAPP_nat_nat(suc,zero_zero_nat))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,X))) = hAPP_nat_real(sin_coeff,X)) & (hBOOL(hAPP_nat_bool(even_odd_even_nat,X)) -> hAPP_nat_real(sin_coeff,X) = zero_zero_real))) # label(fact_3795_sin__coeff__def) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4330 (all N_1 hAPP_int_int(ring_1_of_int_int,hAPP_nat_int(semiri1621563631at_int,N_1)) = hAPP_nat_int(semiri1621563631at_int,N_1)) # label(fact_4579_of__int__int__eq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4331 (all P all Q all R hAPP_P1853363688nt_rat(hAPP_f1065357839nt_rat(hAPP_f1120137083nt_rat(cOMBS_549334880nt_rat,P),Q),R) = hAPP_int_rat(hAPP_P507092031nt_rat(P,R),hAPP_P1175774780nt_int(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oin_002) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4332 (all N_26 hAPP_int_int(number_number_of_int,hAPP_nat_int(semiri1621563631at_int,N_26)) = hAPP_nat_int(semiri1621563631at_int,N_26)) # label(fact_2825_number__of__int) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4333 (all Lx_4 all Ly_2 all Rx_4 all Ry_2 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_4),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ly_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_4),Ry_2))) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_4),Ly_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_4),Ry_2))) # label(fact_153_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4334 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_2)) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) = hAPP_P1175774780nt_int(product_fst_int_int,posDivAlg(A_3,B_2))))) # label(fact_4225_div__pos__pos) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4335 (all A_72 all B_64 all C_32 hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_72),hAPP_Q1762011733de_int(div_mo231679042de_int(B_64),C_32))),C_32) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_72),B_64)),C_32)) # label(fact_2317_mod__mult__right__eq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4336 (all Ta all B_1 all D_1 (is_int(Ta) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ta),B_1)) -> (all X (is_int(X) -> ((all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B_1)) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) != X)))))) -> (Ta != X -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),D_1) != Ta)))))))) # label(fact_5099_bset_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4337 (all W_8 hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,W_8)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_int_real(number267125858f_real,W_8))),hAPP_int_real(number267125858f_real,W_8))) # label(fact_449_number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4338 (all N_1 (N_1 != zero_zero_nat -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)))) # label(fact_1744_gr0I) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4339 (all Lx_2 all Ly all Rx_2 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_2),Ly)),Rx_2) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ly),Rx_2))) # label(fact_165_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4340 (all B_93 all A_114 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_114)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_93)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_114),B_93)))))) # label(fact_1880_add__nonneg__pos) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4341 (all X_2 all Y_2 (hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_2),Y_2))) <-> -hBOOL(hAPP_nat_bool(even_odd_even_nat,Y_2)) & -hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2)) | hBOOL(hAPP_nat_bool(even_odd_even_nat,Y_2)) & hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2)))) # label(fact_3756_even__sum__nat) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4342 (all A (is_int(A) -> (zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),A) <-> zero_zero_int = A))) # label(fact_641_double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4343 (all X_1 -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,X_1)),zero_zero_real))) # label(fact_3831_not__exp__le__zero) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4344 (all P_3 all Q_2 hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,P_3),Q_2)),hAPP_int_int(div_mod_int(P_3),Q_2)) = divmod_int(P_3,Q_2)) # label(fact_4196_divmod__int__mod__div) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4345 (all P all Q all R hAPP_P1860904029l_real(hAPP_r938550413l_real(P,R),hAPP_r1195171167l_real(Q,R)) = hAPP_r1195171167l_real(hAPP_f1736103445l_real(hAPP_f1908301369l_real(cOMBS_902912708l_real,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__prod_Itc__RealDef__Oreal_Mtc__) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4346 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M)))) # label(fact_1727_nat__le__linear) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4347 (all Lx_5 all Ly_3 all Rx_5 all Ry_3 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_5),Ly_3)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_5),Ry_3)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_5),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_5),Ly_3)),Ry_3))) # label(fact_146_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4348 (all B_1_1 all B_2_1 (is_bool(B_2_1) -> is_bool(hAPP_bool_bool(B_1_1,B_2_1)))) # label(gsy_c_hAPP_000tc__HOL__Obool_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4349 (all A_163 hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,one_one_complex),A_163))) # label(fact_1150_one__dvd) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4350 (all Y_2 all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),X_2)) -> comple124823625p_real(hAPP_r1134773055l_bool(ord_gr788844697n_real(Y_2),X_2)) = X_2)) # label(fact_4637_Sup__greaterThanLessThan) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4351 (all X_2 all N (is_int(X_2) -> (zero_zero_int != X_2 | zero_zero_nat = N <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(abs_abs_int,X_2)),N)))))) # label(fact_3446_zero__less__zpower__abs__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4352 (all X_1 all N_1 all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),N_1))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),X_1))))) # label(fact_4173_prime__divexp) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4353 (all D all A all B (one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D),A) & one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D),B) <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A),B)) = one_one_nat)) # label(fact_4356_coprime__mul__eq__nat) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4354 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M)),N_1)))) # label(fact_3062_Suc__leI) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4355 (all X_45 all P_7 all Q_11 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_45),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_11)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_45),P_7)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_45),Q_11))) # label(fact_102_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4356 (all U all Ma all N all J all I_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),I_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U)),N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),J)),U)),Ma)),N))))) # label(fact_1852_nat__le__add__iff1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4357 (all P all Q all R hAPP_P1175774780nt_int(hAPP_i61245889nt_int(hAPP_f578362681nt_int(cOMBC_1417754532nt_int,P),Q),R) = hAPP_int_int(hAPP_P2110489235nt_int(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oin) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4358 (all Z_1 (is_int(Z_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1)) -> -(all M_1 hAPP_nat_int(semiri1621563631at_int,M_1) != Z_1)))) # label(fact_3257_nonneg__eq__int) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4359 (all Y_6 all P_6 all N_10 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_6),N_10)) -> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N_10)),P_6)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),P_6))),Y_6))) # label(fact_3177_lemma__realpow__diff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4360 (all C_75 all A_156 all B_132 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_156),B_132)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_75)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_75),A_156)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_75),B_132)))))) # label(fact_1205_comm__mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4361 (all N_48 all A_199 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_199)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_199),one_one_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_199),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_199),N_48))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_199),N_48)))))) # label(fact_725_power__Suc__less) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4362 (all Y_2 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_real_real(hAPP_n546711566l_real(root,N),Y_2)))))) # label(fact_3923_real__root__ge__1__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4363 (all X_2 (one_one_real = X_2 <-> hAPP_real_real(sqrt,X_2) = one_one_real)) # label(fact_3321_real__sqrt__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4364 (all V_3 hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,one_one_int),hAPP_int_int(number_number_of_int,V_3))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_int_nat(number_number_of_nat,V_3))) # label(fact_3240_one__div__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4365 (all Ma all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)))) # label(fact_2936_of__nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4366 (all Sa all Ta all A all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),B)) -> hBOOL(hAPP_f115998247l_bool(hAPP_P1876494581l_bool(member180897546at_nat,hAPP_P369611207at_nat(hAPP_P402336767at_nat(produc494345619at_nat,hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,A),Sa)),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,B),Ta))),pair_less)))) # label(fact_4540_pair__lessI1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4367 (all X_1 all Y_1 ratreal(hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,ratreal(X_1)),ratreal(Y_1))) # label(fact_4700_real__plus__code) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4368 (all N_1 all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M)) -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N_1)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)))) # label(fact_3128_zdiff__int) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4369 (all X_1 all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),pi)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(cos,X_1)),hAPP_real_real(cos,Y_1))))))) # label(fact_3571_cos__monotone__0__pi) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4370 (all N_26 hAPP_nat_complex(semiri2020571505omplex,N_26) = hAPP_int_complex(number528085621omplex,hAPP_nat_int(semiri1621563631at_int,N_26))) # label(fact_2827_number__of__int) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4371 (all Ma all Ta all K_1 (is_int(K_1) -> (zero_zero_int != K_1 -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),Ta)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),Ma)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),Ta))))))) # label(fact_1805_zdvd__mono) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4372 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(ln,X_1)),zero_zero_real))))) # label(fact_3440_ln__less__zero) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4373 (all B_49 all B_48 all A_57 all C_20 all A_56 (hAPP_int_int(div_mod_int(A_56),C_20) = hAPP_int_int(div_mod_int(A_57),C_20) -> (hAPP_int_int(div_mod_int(B_49),C_20) = hAPP_int_int(div_mod_int(B_48),C_20) -> hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_56),B_48)),C_20) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_57),B_49)),C_20)))) # label(fact_2370_mod__add__cong) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4374 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,one_one_rat),hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A))))) # label(fact_4726_one__le__Fract__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4375 (all A_12 all B_9 (is_int(A_12) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_12),B_9)),B_9)),hAPP_int_int(div_mod_int(A_12),B_9)) = A_12)) # label(fact_2708_mod__div__decomp) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4376 (all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arccos,Y_1)),pi))))) # label(fact_3607_arccos__ubound) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4377 (all N_44 one_one_complex = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,one_one_complex),N_44)) # label(fact_829_power__one) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4378 (all A_236 all B_165 all C_96 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_236),B_165)),C_96) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_236),C_96)),B_165)) # label(fact_215_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4379 (all P_1 all N ((all M_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N)) -> hBOOL(hAPP_nat_bool(P_1,M_1)))) <-> (all X (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))) -> hBOOL(hAPP_nat_bool(P_1,X)))))) # label(fact_4899_all__nat__less__eq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4380 (all X_2 all A (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),hAPP_int_real(real_int,A))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim1246769320r_real(X_2)),A)))) # label(fact_4415_floor__less__eq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4381 (all X_2 (zero_z891286103de_int = X_2 <-> X_2 = zero_z891286103de_int)) # label(fact_1901_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4382 (all A_207 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_207),zero_zero_real) = A_207) # label(fact_632_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4383 (all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_nat_rat(semiri151668891at_rat,N))))) # label(fact_3145_of__nat__0__less__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4384 (all A_11 all N_18 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_11),hAPP_nat_nat(suc,N_18)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_11),N_18)),A_11)) # label(fact_3031_power__Suc2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4385 (all A_209 zero_zero_rat = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_209),zero_zero_rat)) # label(fact_621_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4386 (all Y_2 all X_2 all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),hAPP_i1948725293t_bool(sRStar,P_5))) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),hAPP_i1948725293t_bool(sRStar,P_5))) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(standardRes(P_5),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_2),Y_2))),hAPP_i1948725293t_bool(sRStar,P_5)))))))) # label(fact_2506_SRStar__mult__prop1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4387 (all X_2 (is_int(X_2) -> ((exists Y (hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Y) = X_2 & is_int(Y))) <-> hBOOL(hAPP_int_bool(even_odd_even_int,X_2))))) # label(fact_3222_even__equiv__def) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4388 (all A_47 zero_z126310315umeral = hAPP_C498520661umeral(div_mo1740067990umeral(A_47),one_on1645066479umeral)) # label(fact_2417_mod__by__1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4389 (all Y_1 all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Y_1),X_1)) -> hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),Y_1)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(abs_abs_int,X_1)),hAPP_int_int(abs_abs_int,Y_1)))) # label(fact_3429_abs__div) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4390 (all A_172 all B_147 all C_85 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_172),B_147)),C_85)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,B_147),C_85)))) # label(fact_1089_dvd__mult__right) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4391 (all P_1 all L ((exists X hBOOL(hAPP_real_bool(P_1,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,L),X)))) <-> (exists X (hBOOL(hAPP_real_bool(P_1,X)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,L),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),zero_zero_real))))))) # label(fact_1807_unity__coeff__ex) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4392 (all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,N)),zero_zero_int)) <-> N = zero_zero_nat)) # label(fact_3105_int__le__0__conv) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4393 (all P all Q all R hAPP_f2002277285nt_int(P,hAPP_i1584592887nt_int(Q,R)) = hAPP_i998473677nt_int(hAPP_f142232355nt_int(hAPP_f1377453843nt_int(cOMBB_320287386nt_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__prod_Itc__Int__Oint_Mtc__Int) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4394 (all Z_1 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(norm_norm_complex,Z_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Z_1),hAPP_complex_complex(cnj,Z_1)))) # label(fact_4062_complex__mod__mult__cnj) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4395 (all A all B all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B))))) # label(fact_1568_zcong__def) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4396 (all A_3 all B_2 all X_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),X_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),X_1)))) # label(fact_4323_gcd__semilattice__nat_Ole__infI2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4397 (all A_97 all C_40 all B_80 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_97),C_40)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_80),C_40))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_97),B_80)))) # label(fact_2043_add__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4398 (all I_1 all P_1 all P_5 ((all I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),P_5)) -> (hBOOL(hAPP_nat_bool(P_1,I_2)) -> hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,I_2)),P_5)))))) -> (hBOOL(hAPP_nat_bool(P_1,I_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),P_5)) -> hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)))))) # label(fact_3248_mod__induct__0) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4399 (all N all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N)),Ma)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),Ma)))) # label(fact_3066_less__eq__Suc__le) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4400 (all D all Sa exists Z all X (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Z),X)) -> (-hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X),Sa))) <-> -hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X),Sa)))))) # label(fact_2260_pinf_I10_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4401 (all A_3 all B_2 ((-(hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),B_2)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),zero_zero_int))) -> hAPP_P1975530577nt_int(adjust(B_2),posDivAlg(A_3,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_2))) = posDivAlg(A_3,B_2)) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),zero_zero_int)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),B_2)) -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),A_3) = posDivAlg(A_3,B_2)))) # label(fact_2500_posDivAlg_Osimps) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4402 (all X_2 (one_on1645066479umeral = X_2 <-> one_on1645066479umeral = X_2)) # label(fact_1955_one__reorient) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4403 (all L (is_int(L) -> (L = min <-> hAPP_int_int(bit1,L) = min))) # label(fact_925_rel__simps_I43_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4404 (all Z_1 hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(cnj,Z_1)) = hAPP_complex_real(norm_norm_complex,Z_1)) # label(fact_4074_complex__mod__cnj) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4405 (all Z_1 hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_complex_real(re,Z_1)),hAPP_complex_real(norm_norm_complex,Z_1)) = hAPP_complex_real(re,hAPP_complex_complex(sgn_sgn_complex,Z_1))) # label(fact_4128_Re__sgn) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4406 (all P_1 all I_1 all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),K_1)) -> (hBOOL(hAPP_int_bool(P_1,K_1)) -> ((all I_2 (is_int(I_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),K_1)) -> (hBOOL(hAPP_int_bool(P_1,I_2)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,I_2),one_one_int))))))) -> hBOOL(hAPP_int_bool(P_1,I_1)))))) # label(fact_2231_int__le__induct) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4407 (all R_3 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(real_int,archim856651990g_real(R_3))),one_one_real)),R_3))) # label(fact_4457_real__of__int__ceiling__diff__one__le) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4408 (all A (A != one_one_nat <-> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),one_one_nat)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,one_one_nat),A)))) # label(fact_2281_gcd__lcm__complete__lattice__nat_Obot__less) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4409 (all A all C all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),C)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B),C))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)))) # label(fact_2073_add__less__cancel__right) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4410 (all D_3 all A_3 all B_2 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2)) = one_one_nat -> one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),A_3))) # label(fact_4352_coprime__lmult__nat) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4411 (all B_2 all D_3 all A_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),A_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),B_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2)))))) # label(fact_4009_coprime__mul) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4412 (all M all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),M)) # label(fact_1729_nat__mult__commute) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4413 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_real_real(sqrt,X_1))))) # label(fact_3332_real__sqrt__ge__one) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4414 (all P all Q all R hAPP_rat_rat(hAPP_nat_fun_rat_rat(P,R),hAPP_nat_rat(Q,R)) = hAPP_nat_rat(hAPP_f2062659861at_rat(hAPP_f1131305186at_rat(cOMBS_nat_rat_rat,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__Rat__Orat_000tc__Rat__Orat_U) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4415 (all Y_1 all X_1 (hBOOL(hAPP_int_bool(nat_is_nat,X_1)) -> (hBOOL(hAPP_int_bool(nat_is_nat,Y_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_1)))))) # label(fact_5145_transfer__int__nat__gcd__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4416 (all B_118 all A_141 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_141),zero_z126310315umeral)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_118)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_141),B_118)),zero_z126310315umeral))))) # label(fact_1284_mult__neg__pos) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4417 (all A_14 all B_11 all C_4 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_14),B_11)),B_11)),hAPP_nat_nat(div_mod_nat(A_14),B_11))),C_4) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_14),C_4)) # label(fact_2684_div__mod__equality) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4418 (all Z_1 all W hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),W)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(uminus_uminus_int,Z_1)),hAPP_int_int(uminus_uminus_int,W))) # label(fact_3516_zminus__zadd__distrib) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4419 (all R_3 all P_3 all Q_2 (hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2) = quotient_of(R_3) -> one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,P_3),Q_2))) # label(fact_5016_quotient__of__coprime) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4420 (all M all N_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_1)) -> hAPP_int_int(legacy_zgcd(M),N_1) = hAPP_int_int(legacy_zgcd(N_1),hAPP_int_int(div_mod_int(M),N_1)))) # label(fact_4513_zgcd__non__0) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4421 (all A all P_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,setS(A,P_5))))) # label(fact_3369_SetS__finite) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4422 (all N_60 all X_41 all Y_33 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_41),Y_33)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_41),N_60)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_33),N_60))))) # label(fact_586_dvd__power__same) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4423 (all B_2 all B_5 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_5),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_5))))))) # label(fact_2630_zdiv__mono2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4424 (all N_58 ((zero_zero_nat != N_58 -> zero_zero_real = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),N_58)) & (zero_zero_nat = N_58 -> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),N_58) = one_one_real))) # label(fact_596_power__0__left) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4425 (all B_73 all A_84 all C_36 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_36)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_73),A_84)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_73),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_84),C_36)))))) # label(fact_2157_add__increasing2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4426 (all X_3 (hBOOL(vanishes(X_3)) -> hBOOL(vanishes(hAPP_f2062659861at_rat(hAPP_f1082758869at_rat(cOMBB_rat_rat_nat,uminus_uminus_rat),X_3))))) # label(fact_5132_vanishes__minus) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4427 (all X_2 all Y_2 (X_2 = zero_zero_rat & Y_2 = zero_zero_rat <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_2),X_2)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_2),Y_2))),zero_zero_rat)))) # label(fact_733_sum__squares__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4428 (all X_3 all A (hBOOL(hAPP_complex_bool(sums_complex(X_3),A)) -> hBOOL(hAPP_real_bool(sums_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,re),X_3)),hAPP_complex_real(re,A))))) # label(fact_4871_Re_Osums) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4429 (all N_1 all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> (exists R_1 (hAPP_nat_real(hAPP_r474017924t_real(power_power_real,R_1),hAPP_nat_nat(suc,N_1)) = A_3 & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_1)))))) # label(fact_3249_realpow__pos__nth2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4430 (all P all Q (-hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q)) | hBOOL(Q))) # label(help_fconj_3_1_U) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4431 (all Real1 all Real2 zero_zero_nat = complex_size(hAPP_real_complex(hAPP_r265291036omplex(complex,Real1),Real2))) # label(fact_3967_complex_Osize_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4432 (all C_59 all A_125 all B_104 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_59),A_125)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_59),B_104))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_59)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_125),B_104))))) # label(fact_1450_mult__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4433 (all X_1 all Y_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X_1)),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3405_real__sqrt__ge__abs1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4434 (all W (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(number_number_of_int,W))) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(number_number_of_int,W)))) -> negDivAlg(one_one_int,hAPP_int_int(number_number_of_int,W)) = hAPP_P1975530577nt_int(adjust(hAPP_int_int(number_number_of_int,W)),negDivAlg(one_one_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(number_number_of_int,W))))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(number_number_of_int,W)))) -> negDivAlg(one_one_int,hAPP_int_int(number_number_of_int,W)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(number_number_of_int,W)))))) # label(fact_2295_negDivAlg__eqn__1__number__of) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4435 (all B_12 all A_15 all C_5 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_15),C_5) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_12),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_15),B_12))),hAPP_Q1762011733de_int(div_mo231679042de_int(A_15),B_12))),C_5)) # label(fact_2682_div__mod__equality2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4436 (all N_1 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N_1)),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_1)))) -> N_1 = hAPP_int_nat(nat,archim1246769320r_real(X_1))))) # label(fact_4031_floor__eq3) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4437 (all X_1 hAPP_int_int(abs_abs_int,X_1) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,zero_zero_int),X_1)) # label(fact_5003_gcd__0__left__int) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4438 (all N_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),hAPP_nat_nat(fact_fact_nat,N_1)))) # label(fact_3809_fact__ge__one__nat) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4439 (all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),Y_1)) -> (exists X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_1),one_one_real))) & hAPP_real_real(exp_real,X) = Y_1 & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)))))) # label(fact_3857_lemma__exp__total) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4440 (all X_1 all N_1 (-hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1)) -> X_1 = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),N_1))) # label(fact_3899_odd__real__root__pow) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4441 (all P all Q all R hAPP_f605995867t_bool(P,hAPP_r1134773055l_bool(Q,R)) = hAPP_r279757445t_bool(hAPP_f1840085295t_bool(hAPP_f1083049571t_bool(cOMBB_2050079537l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__HOL__Obool_J_000tc__fun_030) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4442 (all W_5 ((hAPP_int_nat(number_number_of_nat,W_5) != zero_zero_nat -> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),hAPP_int_nat(number_number_of_nat,W_5)) = zero_zero_real) & (hAPP_int_nat(number_number_of_nat,W_5) = zero_zero_nat -> one_one_real = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),hAPP_int_nat(number_number_of_nat,W_5))))) # label(fact_1481_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4443 (all Ma all N (hAPP_nat_nat(suc,zero_zero_nat) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),N) <-> Ma = zero_zero_nat & N = hAPP_nat_nat(suc,zero_zero_nat) | hAPP_nat_nat(suc,zero_zero_nat) = Ma & zero_zero_nat = N)) # label(fact_3054_add__is__1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4444 (all X_2 all A_1 all Ma (hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,A_1),rsetR(Ma))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma)) -> (hAPP_int_int(legacy_zgcd(X_2),Ma) = one_one_int -> hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,image_int_int(hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,times_times_int),X_2),A_1)),rsetR(Ma))))))) # label(fact_4787_RsetR__zmult__mono) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4445 (all B_2 all A_3 (is_int(A_3) & is_int(B_2) -> (B_2 != zero_zero_int | zero_zero_int != A_3 -> (exists C_3 exists D_2 (B_2 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(legacy_zgcd(A_3),B_2)),D_2) & hAPP_int_int(legacy_zgcd(C_3),D_2) = one_one_int & A_3 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(legacy_zgcd(A_3),B_2)),C_3) & is_int(D_2) & is_int(C_3)))))) # label(fact_4538_make__zrelprime) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4446 (all P_2 all P_1 all X_2 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) -> (hBOOL(P_1) <-> hBOOL(P_2))) -> (hBOOL(P_2) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) & hBOOL(P_1)))) # label(fact_1749_conj__le__cong) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4447 (all C_79 all D_15 all A_160 all B_136 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_160),B_136)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_79),D_15)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_136)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_79)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_160),C_79)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_136),D_15)))))))) # label(fact_1187_mult__mono) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4448 (all A_3 all X_1 hAPP_real_real(scaleR_scaleR_real(A_3),X_1) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_3),X_1)) # label(fact_5106_real__scaleR__def) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4449 (all L all U hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_at238088361st_nat(L),U)))) # label(fact_5065_finite__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4450 (all X_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),hAPP_real_real(cos,X_1)))) # label(fact_3566_cos__ge__minus__one) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4451 (all B_118 all A_141 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_141),zero_z891286103de_int)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_118)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_141),B_118)),zero_z891286103de_int))))) # label(fact_1285_mult__neg__pos) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4452 (all X hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,plus_plus_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),sin)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),cos)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X),zero_zero_real))) # label(fact_4792_DERIV__sin__circle__all__zero) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4453 (all L_2 all K_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),L_2)) -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,K_2),L_2)),hAPP_int_int(div_mod_int(K_2),L_2)) = posDivAlg(K_2,L_2)))) # label(fact_2697_posDivAlg__div__mod) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4454 (all A_201 all N_50 all N_49 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_50),N_49)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_201)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_201),one_one_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_201),N_49)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_201),N_50))))))) # label(fact_686_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4455 (all X_1 all N_1 (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,N_1))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> natceiling(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_int_real(number267125858f_real,N_1))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natceiling(X_1)),hAPP_int_nat(number_number_of_nat,N_1))))) # label(fact_3377_natceiling__add__number__of) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4456 (all A all E_2 all C all B all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),E_2)),C)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),E_2)),D))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B)),E_2)),C)),D)))) # label(fact_1572_le__add__iff1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4457 (all Ma all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,Ma)),hAPP_nat_nat(suc,N))))) # label(fact_2859_Suc__less__eq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4458 (all A_231 all N_68 hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_231),N_68)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_231),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_68))) # label(fact_278_power__even__eq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4459 (all A_174 all B_148 all C_86 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_174),B_148)),C_86) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_174),C_86)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_148),C_86))) # label(fact_1072_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4460 (all Y_1 all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(nat,X_1)),hAPP_int_nat(nat,Y_1)) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_1))))) # label(fact_2754_Nat__Transfer_Otransfer__nat__int__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4461 (all Y_1 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Y_1)) -> natfloor(hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_1),hAPP_nat_real(real_nat,Y_1))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,natfloor(X_1)),Y_1)))) # label(fact_3723_natfloor__div__nat) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4462 (all M all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1)),N_1) = M) # label(fact_1648_diff__add__inverse2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4463 (all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K_1)),pls)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),pls)))) # label(fact_269_rel__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4464 (all A_219 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_219),A_219)),A_219) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_219),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) # label(fact_483_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4465 (all N_1 zero_zero_nat = hAPP_int_nat(nat,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,N_1)))) # label(fact_3540_nat__zminus__int) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4466 (all A_25 all B_22 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_25),B_22)) -> B_22 = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,B_22),A_25)),A_25))) # label(fact_2598_dvd__div__mult__self) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4467 (all B_14 all A_17 (is_int(A_17) -> A_17 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_14),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_17),B_14))),hAPP_int_int(div_mod_int(A_17),B_14)))) # label(fact_2671_mod__div__equality2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4468 (all A_124 all C_58 all B_103 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_124),C_58)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_103),C_58))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_58)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_124),B_103))))) # label(fact_1457_mult__right__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4469 (all A_110 all B_89 all C_49 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_110),C_49) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_110),B_89) -> B_89 = C_49)) # label(fact_1919_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4470 (all X_2 all N (zero_zero_nat = N | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),X_2)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_2),N))))) # label(fact_920_nat__zero__less__power__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4471 (all X_1 all Y_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) -> Y_1 != X_1)) # label(fact_1711_dvd_Oless__imp__not__eq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4472 (all A all Ma hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(bnorRset(A),Ma)))) # label(fact_4293_Bnor__fin) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4473 (all K_2 all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),I)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),J_2)))))) # label(fact_1799_mult__less__mono2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4474 (all A_237 all B_166 all C_97 all D_20 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_237),B_166)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_97),D_20)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_237),C_97)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_166),D_20))) # label(fact_196_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4475 (all A all E_2 all C all B all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),E_2)),C)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),E_2)),D))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B)),E_2)),C)),D)))) # label(fact_1578_less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4476 (all A_139 all B_116 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_139),B_116))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_139)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_116))))) # label(fact_1300_zero__less__mult__pos) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4477 (all P all Q all R hAPP_i1948725293t_bool(hAPP_f1791153283t_bool(hAPP_f735247205t_bool(cOMBC_983868720t_bool,P),Q),R) = hAPP_f429935748t_bool(hAPP_i872162435t_bool(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_013) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4478 (all A_28 (is_int(A_28) -> (zero_zero_int != A_28 -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_28),A_28) = one_one_int))) # label(fact_2585_div__self) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4479 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1)))))) # label(fact_3129_n__less__m__mult__n) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4480 (all P_3 all Q_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Q_2),zero_zero_int)) -> normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2)) = normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(uminus_uminus_int,P_3)),hAPP_int_int(uminus_uminus_int,Q_2))))) # label(fact_4651_normalize__negative) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4481 (all A_3 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))) -> one_one_int != hAPP_int_int(inv(P_3),A_3))))) # label(fact_2129_inv__not__1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4482 (all X_1 all Y_1 hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_1)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_1)) # label(fact_4987_abs__gcd__int) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4483 (all X_2 (X_2 = one_one_real <-> one_one_real = X_2)) # label(fact_1954_one__reorient) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4484 (all X_1 hAPP_real_real(abs_abs_real,hAPP_nat_real(real_nat,X_1)) = hAPP_nat_real(real_nat,X_1)) # label(fact_3634_abs__real__of__nat__cancel) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4485 (all N_1 -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),zero_zero_nat))) # label(fact_1634_less__zeroE) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4486 (all A all B all C (C != zero_zero_real -> (hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),C) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),C) <-> B = A))) # label(fact_1598_real__mult__right__cancel) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4487 (all K_2 all M all N_1 hAPP_int_int(legacy_zgcd(hAPP_int_int(legacy_zgcd(K_2),M)),N_1) = hAPP_int_int(legacy_zgcd(K_2),hAPP_int_int(legacy_zgcd(M),N_1))) # label(fact_4490_zgcd__assoc) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4488 (all V all N (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))) & (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))) -> N = hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))) <-> hAPP_int_nat(number_number_of_nat,V) = hAPP_nat_nat(suc,N))) # label(fact_4778_eq__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4489 (all V_20 all V_19 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_19)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_20)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_19)),hAPP_int_nat(number_number_of_nat,V_20)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_19),V_20))))) # label(fact_373_semiring__add__number__of) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4490 (all X_2 (zero_zero_real = hAPP_real_real(sin,X_2) <-> (exists N_2 (hBOOL(hAPP_nat_bool(even_odd_even_nat,N_2)) & X_2 = hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) | (exists N_2 (hBOOL(hAPP_nat_bool(even_odd_even_nat,N_2)) & X_2 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3752_sin__zero__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4491 (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(ln,X_2))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_2))))) # label(fact_3443_ln__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4492 (all A_119 all B_98 all V_6 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_119),B_98)),hAPP_int_int(number_number_of_int,V_6)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_119),hAPP_int_int(number_number_of_int,V_6))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_98),hAPP_int_int(number_number_of_int,V_6)))) # label(fact_1559_left__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4493 (all A_10 all N_17 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_10),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_10),N_17)) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_10),hAPP_nat_nat(suc,N_17))) # label(fact_3034_power__Suc) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4494 (all K_2 all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),I)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),J_2))))) # label(fact_1790_mult__le__mono2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4495 (all X_1 hAPP_complex_complex(sgn_sgn_complex,X_1) = hAPP_complex_complex(scaleR1652505878omplex(hAPP_real_real(inverse_inverse_real,hAPP_complex_real(norm_norm_complex,X_1))),X_1)) # label(fact_5113_complex__sgn__def) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4496 (all M_11 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_11)),zero_zero_int))) # label(fact_2923_of__nat__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4497 (all A all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma)) -> (one_one_int = hAPP_int_int(legacy_zgcd(A),Ma) -> (exists X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),X),Ma)) & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X),hAPP_i1948725293t_bool(norRRset,Ma))) & (all Y (is_int(Y) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y),hAPP_i1948725293t_bool(norRRset,Ma))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),Y),Ma)) -> X = Y))) & is_int(X)))))) # label(fact_4585_norR__mem__unique) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4498 (all Lx_2 all Ly all Rx_2 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_2),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Ly),Rx_2)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_2),Ly)),Rx_2)) # label(fact_164_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4499 (all A_1 all Ma (hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,A_1),rsetR(Ma))) -> hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_1)))) # label(fact_4594_RsetR__fin) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4500 (all M all K_2 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M)) -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1) = K_2 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),K_2))))))) # label(fact_3134_prod__mn__less__k) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4501 (all P_3 hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_P1175774780nt_int(product_snd_int_int,P_3)) = frct(P_3)) # label(fact_4711_Frct__def) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4502 (all D all B all C all A (zero_z126310315umeral != A -> (zero_z126310315umeral != C -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A),B)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,C),D)) -> (hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,B),A) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,D),C) <-> hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A),D) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B),C))))))) # label(fact_2659_dvd__div__div__eq__mult) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4503 (all A all B (hAPP_int_int(legacy_zgcd(A),B) = one_one_int <-> -(exists P_4 (is_int(P_4) & hBOOL(hAPP_int_bool(zprime,P_4)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_4),B)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_4),A)))))) # label(fact_4521_zgcd1__iff__no__common__primedivisor) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4504 (all X_2 all Y_2 all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),B)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),Y_2)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B),X_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B),Y_2)))))) # label(fact_535_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4505 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (exists N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_2)))),X_1))))) # label(fact_3858_reals__Archimedean) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4506 (all N_48 all A_199 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_199)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_199),one_one_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_199),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_199),N_48))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_199),N_48)))))) # label(fact_720_power__Suc__less) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4507 (all Ma all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(suc,Ma))) <-> Ma = N))) # label(fact_3061_le__less__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4508 (all M all K_2 all N_1 hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),N_1)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N_1)) # label(fact_4969_gcd__add__mult__int) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4509 (all V_23 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_23)),one_one_rat) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_23),hAPP_int_int(bit1,pls)))) # label(fact_57_add__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4510 (all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K_1)),pls)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),pls)))) # label(fact_266_rel__simps_I12_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4511 (all X_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)))))) # label(fact_3915_real__root__ge__zero) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4512 (all X_3 ((all J_1 exists M_3 all M_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_3),M_1)) -> (all N_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_3),N_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_nat_real(X_3,M_1)),hAPP_nat_real(X_3,N_2)))),hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,J_1))))))))) <-> hBOOL(cauchy_real(X_3)))) # label(fact_4374_Cauchy__iff2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4513 (all X_1 all N_1 all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> hAPP_real_real(hAPP_n546711566l_real(root,M),hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)) = hAPP_real_real(hAPP_n546711566l_real(root,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1)),X_1)))) # label(fact_3907_real__root__mult__exp) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4514 (all K_1 all Ma all N (Ma = N <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K_1)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K_1)),Ma))) # label(fact_2878_Suc__mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4515 (all P all Q all R hAPP_i1948725293t_bool(hAPP_f720654810t_bool(hAPP_f908327869t_bool(cOMBB_1290694363ol_int,P),Q),R) = hAPP_f429935748t_bool(P,hAPP_int_fun_int_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4516 (all A_11 all N_18 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_11),hAPP_nat_nat(suc,N_18)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_11),N_18)),A_11)) # label(fact_3027_power__Suc2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4517 (all V_20 all V_19 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_19)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_20)) -> hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_19),V_20)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_19)),hAPP_int_complex(number528085621omplex,V_20))))) # label(fact_375_semiring__add__number__of) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4518 (all X_1 all Y_1 all N_1 (zero_zero_nat != N_1 -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),N_1)),Y_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))))) # label(fact_1592_divides__exp2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4519 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(sqrt,X_1))))) # label(fact_3326_real__sqrt__gt__zero) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4520 (all N_57 all A_211 all B_157 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_211),B_157)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_211)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_57)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_211),N_57)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_157),N_57))))))) # label(fact_602_power__strict__mono) # label(axiom) # label(non_clause). [assumption]. 5.74/5.74 4521 (all V_15 all W_13 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_15)),hAPP_int_rat(number_number_of_rat,W_13)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_15),W_13))) # label(fact_428_arith__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4522 (all A_105 A_105 = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,zero_z891286103de_int),A_105)) # label(fact_1985_add__0) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4523 (all K_2 all I all J_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,I),J_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,J_2),K_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,I),K_2))))) # label(fact_1521_real__le__trans) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4524 (all C_1 all A_3 all B_2 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),A_3)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2)) -> (B_2 = C_1 -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C_1),A_3)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),C_1))))) # label(fact_1714_dvd_Oord__less__eq__trans) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4525 (all A_10 all N_17 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_10),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_10),N_17)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_10),hAPP_nat_nat(suc,N_17))) # label(fact_3036_power__Suc) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4526 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),one_one_real)) -> hAPP_real_real(uminus_uminus_real,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),X_1))))) = hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),X_1)))) # label(fact_3498_aux5) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4527 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),one_one_real)) -> hAPP_real_real(cos,hAPP_real_real(arcsin,X_1)) = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3558_cos__arcsin) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4528 (all A all C all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),B)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),C)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B),C))))) # label(fact_2031_add__le__cancel__right) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4529 (all P all Q (is_bool(P) -> hAPP_nat_bool(cOMBK_bool_nat(P),Q) = P)) # label(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4530 (all A_93 (is_int(A_93) -> A_93 = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_93),zero_zero_int))) # label(fact_2078_diff__0__right) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4531 (all K_2 (is_int(K_2) -> hAPP_int_int(number_number_of_int,K_2) = K_2)) # label(fact_262_number__of__is__id) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4532 (all P_1 P_1 = collect_nat(P_1)) # label(fact_207_Collect__def) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4533 (all M_8 all N_22 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(semiri132038758t_real,M_8)),hAPP_nat_real(semiri132038758t_real,N_22)) = hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_8),N_22))) # label(fact_2959_of__nat__add) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4534 (all X_49 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_49),X_49) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_49),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_31_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4535 (all M all N_1 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N_1))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),N_1)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),M))))) # label(fact_1400_zprime__zdvd__zmult__better) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4536 (all M all N_1 (N_1 = M | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)))) # label(fact_1767_less__or__eq__imp__le) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4537 (all X_49 hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_49),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,X_49),X_49)) # label(fact_29_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4538 (all K_2 all A_3 all J_2 all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),zero_zero_int),P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(multInv(P_3),J_2)),J_2)),K_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(multInv(P_3),J_2)),A_3)),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(K_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),J_2))),P_3))))))) # label(fact_2220_aux______4) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4539 (all M all K_2 (M != hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K_2) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K_2))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),K_2))))) # label(fact_3133_one__less__k) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4540 (all Y_1 (Y_1 = hAPP_real_real(tan,hAPP_real_real(arctan,Y_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arctan,Y_1)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arctan,Y_1))))) # label(fact_3622_arctan) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4541 (all X_1 hAPP_real_real(sin,hAPP_real_real(uminus_uminus_real,X_1)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_1))) # label(fact_3633_sin__minus) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4542 (all P_1 all A0 all A1 (hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(norm_frac_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A0),A1))) -> ((all A_2 all B_4 (is_int(B_4) & is_int(A_2) -> (hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(norm_frac_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_2),B_4))) -> ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_4),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,hAPP_int_int(uminus_uminus_int,A_2)),hAPP_int_int(uminus_uminus_int,B_4)))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A_2),B_4)))))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A0),A1))))) # label(fact_4571_norm__frac_Opinduct) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4543 (all A_109 all B_88 all C_48 (hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_109),C_48) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_109),B_88) -> B_88 = C_48)) # label(fact_1929_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4544 (all A_3 all C_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,A_3)),C_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_3),C_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,zero_zero_nat)),C_1))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(div_mod_nat(A_3),C_1)),hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,zero_zero_nat)),C_1))),C_1))) # label(fact_3188_div__Suc) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4545 (all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2)) -> zero_zero_nat = B_2 | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_3),B_2)))) # label(fact_1611_divides__ge) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4546 (all A_227 (is_int(A_227) -> A_227 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,one_one_int),A_227))) # label(fact_334_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4547 (all Y_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arctan,Y_1)))) # label(fact_3502_arctan__lbound) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4548 (all Y_1 all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1)) -> hAPP_nat_nat(div_mod_nat(hAPP_int_nat(nat,X_1)),hAPP_int_nat(nat,Y_1)) = hAPP_int_nat(nat,hAPP_int_int(div_mod_int(X_1),Y_1))))) # label(fact_2763_nat__mod__distrib) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4549 (all A_208 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A_208) = A_208) # label(fact_628_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4550 (all M all N_1 hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N_1)),hAPP_nat_nat(div_mod_nat(M),N_1))))) # label(fact_4219_divmod__nat__rel) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4551 (all V_2 all W hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,W))) = hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,V_2))),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W)))) # label(fact_2461_zmod__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4552 (all W (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,W)),zero_zero_int)) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,one_one_int),hAPP_int_int(number_number_of_int,W)) = hAPP_P1175774780nt_int(product_fst_int_int,hAPP_P1975530577nt_int(negateSnd,negDivAlg(hAPP_int_int(uminus_uminus_int,one_one_int),hAPP_int_int(uminus_uminus_int,hAPP_int_int(number_number_of_int,W))))))) # label(fact_4222_div__pos__neg__1__number__of) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4553 (all A_171 all B_146 all C_84 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_171),B_146)),C_84)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_171),C_84)))) # label(fact_1092_dvd__mult__left) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4554 (all A_232 all C_92 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_232),C_92) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_92),A_232)) # label(fact_243_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4555 (all V_12 all W_10 hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_12),W_10)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_12)),hAPP_int_int(number_number_of_int,W_10))) # label(fact_437_add__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4556 -(all T (is_int(T) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,s),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),one_one_int) != hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)),T))) # label(fact_62__096_B_Bthesis_O_A_I_B_Bt_O_As_A_094_A2_A_L_A1_A_061_A_I4_A_K_Am_A_L_A1_) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4557 (all X_1 all Y_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Y_1)) -> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_1) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Y_1),hAPP_int_int(div_mod_int(X_1),Y_1)))) # label(fact_5006_gcd__non__0__int) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4558 (all Y_1 all X_1 (hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = X_1 -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_1)) -> Y_1 = hAPP_real_real(sqrt,X_1)))) # label(fact_3340_real__sqrt__unique) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4559 (all A all C all B (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A),B)) <-> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A),C)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B),C))))) # label(fact_2030_add__le__cancel__right) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4560 (all P_5 quotient_of(hAPP_rat_rat(uminus_uminus_rat,P_5)) = hAPP_P1975530577nt_int(produc1518849193nt_int(hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),uminus_uminus_int)),quotient_of(P_5))) # label(fact_4833_rat__uminus__code) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4561 (all V_17 all B_160 all C_90 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_i769753017umeral(number1443263063umeral,V_17)),B_160)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_i769753017umeral(number1443263063umeral,V_17)),C_90)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_i769753017umeral(number1443263063umeral,V_17)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_160),C_90))) # label(fact_385_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4562 (all K_1 all L (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K_1)),hAPP_int_int(bit0,L))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L)))) # label(fact_114_rel__simps_I14_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4563 (all X_1 all Y_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Y_1)),X_1))) # label(fact_4330_gcd__semilattice__nat_Oinf__le1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4564 (all F all A_1 (hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1)) -> ((all X (is_int(X) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(F,X)))))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_f1594865479ol_int(hAPP_f1926459811ol_int(big_co230513141nt_int,F),A_1)))))) # label(fact_4944_transfer__nat__int__sum__prod__closure_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4565 (all A_3 all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),P_3)) -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(nat,P_3)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat,P_3)),one_one_nat))))) # label(fact_2766_Euler_Oaux__1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4566 (all A_3 all C_1 all D_3 all B_2 (is_int(D_3) & is_int(B_2) -> (B_2 != zero_zero_int -> (zero_zero_int != D_3 -> hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),D_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_1),B_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),D_3)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)),hAPP_int_rat(hAPP_int_fun_int_rat(fract,C_1),D_3)))))) # label(fact_4719_diff__rat) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4567 (all N_1 all P_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),P_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_3),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(fact,N_1)))))) # label(fact_4551_divides__fact) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4568 (all Z_3 all Z_2 (is_int(Z_2) & is_int(Z_3) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_3)) -> (hAPP_int_nat(nat,Z_2) = hAPP_int_nat(nat,Z_3) <-> Z_2 = Z_3))))) # label(fact_2728_eq__nat__nat__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4569 (all X_1 all N_5 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_5)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N_5),X_1)),hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1))))))) # label(fact_3924_real__root__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4570 (all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) -> N_1 = hAPP_nat_nat(suc,zero_zero_nat) | N_1 = zero_zero_nat)) # label(fact_3226_less__2__cases) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4571 (all A_30 all B_26 all C_11 (is_int(C_11) -> (C_11 != zero_zero_int -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_30),C_11)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_26),C_11)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_30),B_26)))) # label(fact_2577_div__mult__mult2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4572 (all A all B all C (is_int(B) & is_int(C) -> (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),C) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),B) <-> C = B))) # label(fact_1939_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4573 (all A_108 all B_87 all C_47 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_108),B_87)),C_47) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_108),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_87),C_47))) # label(fact_1948_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4574 (all Ma all N (hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Ma),N) = zero_zero_nat <-> Ma = zero_zero_nat & N = zero_zero_nat)) # label(fact_4339_gcd__zero__nat) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4575 (all A_128 all C_62 all B_107 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_128),C_62)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_107),C_62))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_62)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_128),B_107))))) # label(fact_1435_mult__right__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4576 (all P_3 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_2775_Euler_Oaux____2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4577 (all M all N_1 hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,hAPP_nat_nat(div_mod_nat(M),N_1))),N_1) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,M)),N_1)) # label(fact_2881_mod__Suc__eq__Suc__mod) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4578 (all A all B (is_int(B) & is_int(A) -> (hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(norm_frac_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A),B))) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),zero_zero_int)) -> (zero_zero_int = A | B = zero_zero_int -> norm_frac(A,B) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int)) & (-(A = zero_zero_int | B = zero_zero_int) -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A),hAPP_int_int(hAPP_int_fun_int_int(int_gcd,A),B))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B),hAPP_int_int(hAPP_int_fun_int_int(int_gcd,A),B))) = norm_frac(A,B))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),zero_zero_int)) -> norm_frac(A,B) = norm_frac(hAPP_int_int(uminus_uminus_int,A),hAPP_int_int(uminus_uminus_int,B)))))) # label(fact_4803_norm__frac_Opsimps) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4579 (all Z_1 exists A_2 exists R_1 Z_1 = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_1)),expi(A_2))) # label(fact_4051_complex__expi__Ex) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4580 (all X_2 all A_1 (hBOOL(hAPP_P1463840893t_bool(A_1,X_2)) <-> hBOOL(hAPP_f115998247l_bool(hAPP_P1876494581l_bool(member180897546at_nat,X_2),A_1)))) # label(fact_200_mem__def) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4581 (all B_115 all A_138 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_138)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_115),zero_z126310315umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_115),A_138)),zero_z126310315umeral))))) # label(fact_1305_mult__pos__neg2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4582 (all C_81 all A_167 all B_142 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_167),B_142)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_167),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_142),C_81))))) # label(fact_1123_dvd__mult2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4583 (all V all W_1 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_i1732201573de_int(number1226105091de_int,V)),hAPP_i1732201573de_int(number1226105091de_int,W_1))) <-> -hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_i1732201573de_int(number1226105091de_int,W_1)),hAPP_i1732201573de_int(number1226105091de_int,V))))) # label(fact_88_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4584 (all W_1 all X_2 (X_2 = hAPP_i769753017umeral(number1443263063umeral,W_1) <-> hAPP_i769753017umeral(number1443263063umeral,W_1) = X_2)) # label(fact_254_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4585 (all K_2 all M all N_1 hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N_1)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,K_2)),hAPP_int_int(legacy_zgcd(M),N_1))) # label(fact_4502_zgcd__zmult__distrib2__abs) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4586 (all A all B (hAPP_int_int(abs_abs_int,A) = hAPP_int_int(abs_abs_int,B) <-> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3457_power2__eq__iff__abs__eq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4587 (all A_215 all M_24 all N_64 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_24),N_64)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_215),M_24)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_215),N_64))))) # label(fact_559_le__imp__power__dvd) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4588 (all N_55 all A_205 all B_156 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_205),B_156)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_205)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_205),N_55)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_156),N_55)))))) # label(fact_654_power__mono) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4589 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))))) # label(fact_717_pos__zmult__pos) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4590 (all X_39 all Y_31 -hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_39),X_39)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_31),Y_31))),zero_zero_rat))) # label(fact_737_not__sum__squares__lt__zero) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4591 (all Lx_3 all Ly_1 all Rx_3 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_3),Ly_1)),Rx_3) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_3),Rx_3)),Ly_1)) # label(fact_163_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4592 (all B_117 all A_140 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_117),A_140))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_140)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_117))))) # label(fact_1293_zero__less__mult__pos2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4593 (all N all K_1 all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),Ma)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),K_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)))))) # label(fact_1649_le__diff__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4594 (all M all N_1 ((hAPP_nat_nat(suc,hAPP_nat_nat(div_mod_nat(M),N_1)) = N_1 -> hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,M)),N_1) = zero_zero_nat) & (hAPP_nat_nat(suc,hAPP_nat_nat(div_mod_nat(M),N_1)) != N_1 -> hAPP_nat_nat(suc,hAPP_nat_nat(div_mod_nat(M),N_1)) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,M)),N_1)))) # label(fact_3081_mod__Suc) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4595 (all P all Q all R hAPP_i1778081398nt_int(hAPP_f2004554147nt_int(hAPP_f1228585167nt_int(cOMBB_1102189010nt_int,P),Q),R) = hAPP_f1574055885nt_int(P,hAPP_int_fun_int_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc___024) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4596 (all A_3 all Q_2 all R_3 (is_int(A_3) & is_int(Q_2) -> (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,A_3),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3))) -> (zero_zero_int != A_3 -> Q_2 = one_one_int)))) # label(fact_2793_self__quotient) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4597 (all A_3 hAPP_real_real(log(A_3),one_one_real) = zero_zero_real) # label(fact_3862_log__one) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4598 (all P_1 ((all X (is_int(X) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> hBOOL(hAPP_nat_bool(P_1,hAPP_int_nat(nat,X)))))) <-> (all X1 hBOOL(hAPP_nat_bool(P_1,X1))))) # label(fact_2726_all__nat) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4599 (all Ma all N (N = Ma <-> hAPP_nat_rat(semiri151668891at_rat,Ma) = hAPP_nat_rat(semiri151668891at_rat,N))) # label(fact_2836_of__nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4600 (all M_10 all N_24 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_10),N_24)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,M_10)),hAPP_nat_nat(semiri984289939at_nat,N_24))))) # label(fact_2941_less__imp__of__nat__less) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4601 (all A_203 all N_53 all B_155 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_203),N_53)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_155),N_53))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_155)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_203),B_155))))) # label(fact_670_power__less__imp__less__base) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4602 (all A_16 all B_13 (is_int(A_16) -> A_16 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_16),B_13)),B_13)),hAPP_int_int(div_mod_int(A_16),B_13)))) # label(fact_2675_mod__div__equality) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4603 (all A_21 all C_6 all B_18 (zero_z891286103de_int != B_18 -> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_6),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_21),B_18)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_21),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_6),B_18))),B_18))) # label(fact_2650_div__mult__self1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4604 (all X_2 hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),cos)),cos),X_2),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_2))),hAPP_real_real(cos,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_2))),hAPP_real_real(cos,X_2)))))) # label(fact_4776_DERIV__cos__cos__mult) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4605 (all V_17 all B_160 all C_90 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_17)),B_160)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_17)),C_90)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_17)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_160),C_90))) # label(fact_389_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4606 (all A all B all Ma (is_int(B) -> ((exists K (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ma),K)) = B & is_int(K))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma))))) # label(fact_1397_zcong__iff__lin) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4607 (all Z_7 all W_2 all Y_10 all X_14 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_10),X_14)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_7),W_2)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_14),W_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Y_10),Z_7)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X_14),Y_10)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,W_2),Z_7))))) # label(fact_2608_div__mult__div__if__dvd) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4608 (all C_74 all A_155 all B_131 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_155),B_131)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_74)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_74),A_155)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_74),B_131)))))) # label(fact_1211_mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4609 (all X hAPP_nat_real(diffs_real(cos_coeff),X) = hAPP_real_real(uminus_uminus_real,hAPP_nat_real(sin_coeff,X))) # label(fact_4878_cos__fdiffs) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4610 (all A all B all Q_1 all R_2 (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A,B),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_1),R_2))) -> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(hAPP_int_int(uminus_uminus_int,A),B),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,R_2),zero_zero_int)),hAPP_int_int(uminus_uminus_int,Q_1)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(uminus_uminus_int,Q_1)),one_one_int))),hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,R_2),zero_zero_int)),zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B),R_2))))))) # label(fact_3555_zminus1__lemma) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4611 (all W_1 all Z_2 (is_int(Z_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2)) <-> (exists N_2 Z_2 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_1),hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_2))))))) # label(fact_2848_zless__iff__Suc__zadd) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4612 (all Y_2 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_int_rat(number_number_of_rat,Y_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Y_2)))) # label(fact_758_less__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4613 (all Z_1 all W hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),W)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(uminus_uminus_int,Z_1)),W)) # label(fact_3514_zmult__zminus) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4614 (all A_67 all C_27 all B_59 hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_67),B_59)),C_27) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_67),C_27)),B_59)),C_27)) # label(fact_2336_zmod__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4615 (all L all U hAPP_n1699378549t_bool(ord_at4362885an_nat(hAPP_nat_nat(suc,L)),U) = hAPP_n1699378549t_bool(ord_gr660468384an_nat(L),U)) # label(fact_4904_atLeastSucLessThan__greaterThanLessThan) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4616 (all B_138 all A_162 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_138)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_162),zero_zero_nat)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_162)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_138),zero_zero_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_162),B_138)),zero_zero_nat)))) # label(fact_1174_split__mult__neg__le) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4617 (all C_63 all D_10 all A_129 all B_108 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_129),B_108)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_63),D_10)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_129)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_63)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_129),C_63)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_108),D_10)))))))) # label(fact_1429_mult__le__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4618 (all G all F all X_2 all D (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D)) -> ((all Z (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_2))),D)) -> hAPP_real_real(G,hAPP_real_real(F,Z)) = Z)) -> ((all Z (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_2))),D)) -> hBOOL(hAPP_real_bool(isCont_real_real(F),Z)))) -> (exists E (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),E)) & (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y),hAPP_real_real(F,X_2)))),E)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y),hAPP_real_real(F,X_2))))) -> hAPP_real_real(F,hAPP_real_real(G,Y)) = Y)))))))) # label(fact_5179_isCont__inv__fun__inv) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4619 (all X_1 all A_3 all B_2 hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_3),B_2)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),B_2))) # label(fact_4412_powr__divide2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4620 (all K_1 all L hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,K_1),L) = hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,L),zero_zero_int)),K_1),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,L),hAPP_int_int(div_mod_int(hAPP_int_int(abs_abs_int,K_1)),hAPP_int_int(abs_abs_int,L)))))) # label(fact_5014_gcd__code__int) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4621 (all B_117 all A_140 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_117),A_140))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_140)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_117))))) # label(fact_1291_zero__less__mult__pos2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4622 (all X_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(sin,X_1))),one_one_real))) # label(fact_3662_abs__sin__le__one) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4623 (all A_20 all B_17 (zero_z126310315umeral != B_17 -> hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_20),B_17)),one_on1645066479umeral) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_20),B_17)),B_17))) # label(fact_2653_div__add__self2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4624 (all A_176 zero_zero_complex = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,zero_zero_complex),A_176)) # label(fact_1047_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4625 (all A all B all C all X_2 (is_int(A) -> (hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),B) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,C),hAPP_int_nat(nat,X_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_int_nat(nat,X_2))) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zOdd)) -> (hAPP_int_int(legacy_zgcd(A),B) = one_one_int -> (exists K (A = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K),hAPP_int_nat(nat,X_2)) & is_int(K))))))))) # label(fact_4528_int__relprime__odd__power__divisors) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4626 (all X_1 ratreal(hAPP_rat_rat(uminus_uminus_rat,X_1)) = hAPP_real_real(uminus_uminus_real,ratreal(X_1))) # label(fact_4702_real__uminus__code) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4627 (all X_44 all Y_36 all Z_15 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_44),Y_36)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_44),Z_15)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_44),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Y_36),Z_15))) # label(fact_326_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4628 (all M_10 all N_24 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_10),N_24)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(semiri151668891at_rat,M_10)),hAPP_nat_rat(semiri151668891at_rat,N_24))))) # label(fact_2938_less__imp__of__nat__less) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4629 (all Ta all A_1 all D all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),D_1)) -> (all X (is_int(X) -> ((all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A_1)) -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) != X)))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Ta))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),D_1)),Ta))))))))) # label(fact_5104_aset_I9_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4630 (all U_1 all M all N_1 all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),U_1)),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_2),U_1)),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),I)),U_1)),N_1)))) # label(fact_1848_nat__diff__add__eq2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4631 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X_1)))))) # label(fact_3817_aux4) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4632 (all Ma all N (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(semiri1424489471de_int,Ma)),hAPP_n522471361de_int(semiri1424489471de_int,N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)))) # label(fact_2933_of__nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4633 (all M_26 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M_26),M_26) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),one_one_int)),M_26)) # label(fact_406_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4634 (all L all U hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(ord_at875362053st_int(L),U)) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,U),L)),one_one_int))) # label(fact_5081_card__atLeastAtMost__int) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4635 (all A_226 all B_161 all V_18 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_226),B_161)),hAPP_i1732201573de_int(number1226105091de_int,V_18)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_226),hAPP_i1732201573de_int(number1226105091de_int,V_18))),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_161),hAPP_i1732201573de_int(number1226105091de_int,V_18)))) # label(fact_383_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4636 (all N_1 all K_2 all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),M)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N_1)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),K_2))))) # label(fact_1650_Nat_Odiff__diff__eq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4637 (all X_8 all Q_6 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_8),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_8),Q_6)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_8),hAPP_nat_nat(suc,Q_6))) # label(fact_3019_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4638 (all W all V_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_2))) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,W)),hAPP_int_int(number_number_of_int,V_2)))) -> negDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(number_number_of_int,V_2)) = hAPP_P1975530577nt_int(adjust(hAPP_int_int(number_number_of_int,V_2)),negDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(number_number_of_int,V_2))))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,W)),hAPP_int_int(number_number_of_int,V_2)))) -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,W)),hAPP_int_int(number_number_of_int,V_2))) = negDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(number_number_of_int,V_2))))) # label(fact_2497_negDivAlg__eqn__number__of) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4639 (all P all Q all R hAPP_r265291036omplex(P,hAPP_nat_real(Q,R)) = hAPP_n1983812447omplex(hAPP_f23851874omplex(hAPP_f406530075omplex(cOMBB_1714920583ex_nat,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__C) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4640 (all F all X_2 all L (hBOOL(hAPP_real_bool(deriv_real(F,X_2),L)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),L)) -> (exists D_2 ((all H_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H_1),D_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,X_2)),hAPP_real_real(F,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),H_1))))))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2))))))) # label(fact_3888_DERIV__pos__inc__right) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4641 (all C_41 all A_98 all B_81 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_41),A_98)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_41),B_81))) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_98),B_81)))) # label(fact_2040_add__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4642 (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X_2)),one_one_real)) -> hBOOL(monoseq_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,one_one_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_2)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))))) # label(fact_4804_monoseq__arctan__series) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4643 (all A_163 hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,one_on1684967323de_int),A_163))) # label(fact_1151_one__dvd) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4644 (all Z_1 hAPP_int_real(ring_1_of_int_real,Z_1) = hAPP_complex_real(re,hAPP_int_complex(ring_11397209091omplex,Z_1))) # label(fact_4577_complex__Re__of__int) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4645 (all X_1 hAPP_complex_complex(invers1449016382omplex,X_1) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_complex_real(re,X_1)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(im,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,X_1))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(im,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_4109_complex__inverse__def) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4646 (all A_29 all B_25 all C_10 (C_10 != zero_z891286103de_int -> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_10),A_29)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_10),B_25)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_29),B_25))) # label(fact_2584_div__mult__mult1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4647 (all P all Q all R hAPP_P603027463t_bool(hAPP_f892584630t_bool(hAPP_f492612297t_bool(cOMBB_798864284nt_int,P),Q),R) = hAPP_int_bool(P,hAPP_P1175774780nt_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__HOL__Obool_000tc__prod_Itc__Int__Oi) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4648 (all X_46 (is_int(X_46) -> X_46 = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_46),one_one_nat))) # label(fact_79_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4649 (all A_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A_3)) -> zero_zero_nat = A_3)) # label(fact_2274_gcd__lcm__complete__lattice__nat_Otop__le) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4650 (all S (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S)) <-> (exists K hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,S),hAPP_n1699378549t_bool(ord_atMost_nat,K)))))) # label(fact_5194_finite__nat__iff__bounded__le) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4651 (all M all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(div_mod_nat(M),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N_1)),N_1)) # label(fact_2745_div__mod__equality_H) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4652 (all X_45 all P_7 all Q_11 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_45),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_11)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_45),P_7)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_45),Q_11))) # label(fact_101_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4653 (all A_175 all E_3 all B_149 all C_87 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_175),B_149)),E_3)),C_87) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_175),E_3)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_149),E_3)),C_87))) # label(fact_1064_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4654 (all M M = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),zero_zero_nat)) # label(fact_1747_Nat_Oadd__0__right) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4655 (all A_62 all B_54 all C_25 hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_62),hAPP_C498520661umeral(div_mo1740067990umeral(B_54),C_25))),C_25) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_62),B_54)),C_25)) # label(fact_2352_mod__add__right__eq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4656 (all N_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1))))) # label(fact_3766_real__of__nat__fact__ge__zero) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4657 (all A_3 all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(legendre(A_3),P_3)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),P_3))))) # label(fact_2777_Euler__Criterion) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4658 (all B_2 all D_3 all A_3 all C_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),C_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),D_3)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,C_1),D_3)))))) # label(fact_4317_gcd__semilattice__nat_Oinf__mono) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4659 (all D all Sa exists Z all X (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Z)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Sa))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Sa)))))) # label(fact_2256_minf_I9_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4660 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N_1)),M))))) # label(fact_2737_div__less__dividend) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4661 (all N all P_1 (-hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(P_1,N)) -> (exists K (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N)) & (all I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),K)) -> -hBOOL(hAPP_nat_bool(P_1,I_2)))) & hBOOL(hAPP_nat_bool(P_1,K))))))) # label(fact_2243_ex__least__nat__le) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4662 (all U hAPP_f22106695ol_nat(finite_card_nat,hAPP_n1699378549t_bool(ord_lessThan_nat,U)) = U) # label(fact_4950_card__lessThan) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4663 (all A_3 all C_1 hAPP_int_rat(hAPP_int_fun_int_rat(fract,zero_zero_int),A_3) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,zero_zero_int),C_1)) # label(fact_4674_eq__rat_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4664 (all M all N_1 all K_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),K_2))) # label(fact_1721_nat__add__assoc) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4665 (all M hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,M))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3203_div2__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4666 (all U_1 all X_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,U_1),U_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),X_1)))) # label(fact_3481_real__minus__mult__self__le) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4667 (all A all B all C (hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),B) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),C) <-> C = B)) # label(fact_1944_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4668 (all M (hBOOL(hAPP_int_bool(zprime,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),M)),one_one_int))) -> one_one_int = hAPP_int_int(legendre(hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),M)),one_one_int)))) # label(fact_1508_Legendre__1mod4) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4669 (all W_6 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),hAPP_int_rat(number_number_of_rat,W_6))),hAPP_int_rat(number_number_of_rat,W_6)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,W_6))) # label(fact_745_number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4670 (all N_1 all M all K_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2)) -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M) -> M = N_1))) # label(fact_2288_mult__left__cancel) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4671 (all C_33 all D_5 all A_78 all B_67 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_78),B_67)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_33),D_5)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_78),C_33)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_67),D_5)))))) # label(fact_2210_add__less__le__mono) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4672 (all W_5 ((hAPP_int_nat(number_number_of_nat,W_5) = zero_zero_nat -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,zero_zero_int),hAPP_int_nat(number_number_of_nat,W_5)) = one_one_int) & (zero_zero_nat != hAPP_int_nat(number_number_of_nat,W_5) -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,zero_zero_int),hAPP_int_nat(number_number_of_nat,W_5)) = zero_zero_int))) # label(fact_1479_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4673 (all A_227 A_227 = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,one_on1684967323de_int),A_227)) # label(fact_339_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4674 (all M all N_1 hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N_1),divmod_nat(M,N_1)))) # label(fact_4216_divmod__nat__rel__divmod__nat) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4675 (all X_2 ((exists Y hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),Y) = X_2) <-> hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2)))) # label(fact_3768_even__nat__equiv__def2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4676 (all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),one_one_nat)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_2771_Suc__n__div__2__gt__zero) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4677 (all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)) = hAPP_int_nat(phi,P_3))) # label(fact_2781_phi__prime) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4678 (all N_4 ((exists M_1 all X (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),N_4)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),M_1)))) <-> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_4)))) # label(fact_4625_finite__nat__set__iff__bounded) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4679 (all C_60 all A_126 all B_105 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_60),A_126)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_60),B_105))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_60)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_126),B_105))))) # label(fact_1446_mult__left__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4680 (all X_1 all Y_1 hAPP_complex_complex(invers1449016382omplex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,of_real_complex(X_1)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),of_real_complex(Y_1)))) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,of_real_complex(hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),of_real_complex(hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,Y_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))))) # label(fact_3985_complex__inverse__complex__split) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4681 (all M_12 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_nat_rat(semiri151668891at_rat,M_12)))) # label(fact_2912_zero__le__imp__of__nat) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4682 (all X_2 (is_int(X_2) -> (zero_zero_int = X_2 <-> X_2 = zero_zero_int))) # label(fact_1896_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4683 (all B_143 all A_168 all C_82 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_168),C_82)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_168),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_143),C_82))))) # label(fact_1114_dvd__mult) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4684 (all A_192 all M_20 all N_43 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_192),M_20)),N_43) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_192),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_20),N_43))) # label(fact_836_power__mult) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4685 (all A all W_1 (zero_z126310315umeral = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A),hAPP_int_nat(number_number_of_nat,W_1)) <-> zero_z126310315umeral = A & hAPP_int_nat(number_number_of_nat,W_1) != zero_zero_nat)) # label(fact_1385_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4686 (all K_2 all M hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(K_2),K_2),M))) # label(fact_1003_zcong__refl) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4687 (all X_2 all R_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),R_2)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,R_2)),X_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X_2)),R_2)))) # label(fact_3484_abs__le__interval__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4688 (all P P = hAPP_real_real(cOMBI_real,P)) # label(help_COMBI_1_1_COMBI_000tc__RealDef__Oreal_U) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4689 (all K_1 all L (quickcheck_int_of(K_1) = quickcheck_int_of(L) <-> K_1 = L)) # label(fact_4254_Quickcheck__Narrowing_Oint__of__inject) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4690 (all Q_1 all Ma all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),Q_1)),Ma)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),hAPP_nat_nat(suc,Q_1)))) <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,Ma),N) = Q_1))) # label(fact_3213_split__div__lemma) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4691 (all Z_1 hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_complex_real(re,Z_1)),hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,Z_1))) = hAPP_complex_complex(cnj,Z_1)) # label(fact_4131_cnj__def) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4692 (all S_1 all T_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,S_1),T_1)) -> T_1 != S_1)) # label(fact_1688_less__not__refl3) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4693 (all X_1 X_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),X_1)) # label(fact_4303_gcd__idem__nat) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4694 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> hAPP_real_real(inverse_inverse_real,X_1) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,X_1))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3854_real__inv__sqrt__pow2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4695 (all X_2 (hAPP_real_real(cos,X_2) != zero_zero_real -> hBOOL(hAPP_real_bool(deriv_real(tan,X_2),hAPP_real_real(inverse_inverse_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X_2)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3877_DERIV__tan) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4696 (all N_44 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,one_on1684967323de_int),N_44) = one_on1684967323de_int) # label(fact_830_power__one) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4697 (all M all N_1 hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N_1) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M),N_1)),N_1)) # label(fact_4970_gcd__add1__int) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4698 (all A all W_1 (zero_zero_nat != hAPP_int_nat(number_number_of_nat,W_1) & A = zero_zero_rat <-> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A),hAPP_int_nat(number_number_of_nat,W_1)) = zero_zero_rat)) # label(fact_1388_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4699 (all A all B all N (N != zero_zero_nat -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B),N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B))))) # label(fact_2285_pow__divides__eq__nat) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4700 (all Ma all D (is_int(Ma) -> (zero_zero_int = hAPP_int_int(div_mod_int(Ma),D) <-> (exists Q_4 (hAPP_int_int(hAPP_int_fun_int_int(times_times_int,D),Q_4) = Ma & is_int(Q_4)))))) # label(fact_2439_zmod__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4701 (all C_46 all A_103 all B_86 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_46),A_103)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_46),B_86))) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_103),B_86)))) # label(fact_1998_add__le__imp__le__left) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4702 (all Y_2 all X hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,plus_plus_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,minus_minus_real),hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,sin),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,plus_plus_real),Y_2)))),hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,plus_plus_real),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),sin)),hAPP_real_real(cos,Y_2)))),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),cos)),hAPP_real_real(sin,Y_2)))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,minus_minus_real),hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,cos),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,plus_plus_real),Y_2)))),hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,minus_minus_real),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),cos)),hAPP_real_real(cos,Y_2)))),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),sin)),hAPP_real_real(sin,Y_2)))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X),zero_zero_real))) # label(fact_4800_lemma__DERIV__sin__cos__add) # label(axiom) # label(non_clause). [assumption]. 5.74/5.75 4703 (all A_3 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),A_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(zfact,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),P_3))))))) # label(fact_2712_zfact__prop) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4704 (all A all W_1 (is_int(A) -> (zero_zero_int = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),hAPP_int_nat(number_number_of_nat,W_1)) <-> zero_zero_int = A & hAPP_int_nat(number_number_of_nat,W_1) != zero_zero_nat))) # label(fact_1382_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4705 (all R_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),zero_zero_real)) -> (exists N_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N_2)),R_3)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N_2),one_one_int)))) & is_int(N_2))))) # label(fact_4476_reals__Archimedean__6c__int) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4706 (all I all M hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),M))))) # label(fact_3058_less__add__Suc1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4707 (all B_70 all A_81 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_81),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_70),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_81),B_70)),zero_zero_int))))) # label(fact_2185_add__neg__neg) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4708 (all X_1 all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> (A_3 != one_one_real -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> hAPP_real_real(uminus_uminus_real,hAPP_real_real(log(A_3),X_1)) = hAPP_real_real(log(A_3),hAPP_real_real(inverse_inverse_real,X_1)))))) # label(fact_3874_log__inverse) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4709 (all Y_9 all Z_6 all X_13 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,Z_6),X_13)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,Z_6),Y_9)) -> hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X_13),Y_9)),Z_6) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,X_13),Z_6)),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,Y_9),Z_6))))) # label(fact_2609_div__add) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4710 (all Z_1 all W hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),W) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W),Z_1)) # label(fact_265_zadd__commute) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4711 (all N_1 hAPP_int_nat(number_number_of_nat,N_1) = natfloor(hAPP_int_real(number267125858f_real,N_1))) # label(fact_3386_natfloor__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4712 (all N_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1))))) # label(fact_3767_real__of__nat__fact__gt__zero) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4713 (all A_124 all C_58 all B_103 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_124),C_58)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_103),C_58))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_58)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_124),B_103))))) # label(fact_1456_mult__right__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4714 (all M hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),one_one_nat) = one_one_nat) # label(fact_4335_gcd__1__nat) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4715 (all W_1 all Ma (is_int(W_1) -> ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),W_1)) -> Ma = zero_zero_nat) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),W_1)) -> hAPP_nat_int(semiri1621563631at_int,Ma) = W_1) <-> hAPP_int_nat(nat,W_1) = Ma))) # label(fact_3169_nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4716 (all A_3 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> (exists R_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_1)) & hAPP_nat_real(hAPP_r474017924t_real(power_power_real,R_1),N_1) = A_3))))) # label(fact_1628_realpow__pos__nth) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4717 (all C_43 all A_100 all B_83 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_100),B_83)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_43),A_100)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_43),B_83))))) # label(fact_2016_add__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4718 (all C_77 all B_134 all A_158 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_134),A_158)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_77),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_77),A_158)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_77),B_134)))))) # label(fact_1194_mult__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4719 (all M_17 all N_30 ((N_30 != zero_zero_nat -> hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,M_17),N_30) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,M_17),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,M_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_30),one_one_nat)))) & (zero_zero_nat = N_30 -> one_on1684967323de_int = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,M_17),N_30)))) # label(fact_1834_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4720 (all N_63 all M_23 all X_42 all Y_34 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_42),Y_34)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_63),M_23)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_42),N_63)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_34),M_23)))))) # label(fact_565_dvd__power__le) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4721 (all A_34 hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_34),one_on1645066479umeral) = A_34) # label(fact_2547_div__by__1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4722 (all B_143 all A_168 all C_82 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_168),C_82)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_168),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_143),C_82))))) # label(fact_1118_dvd__mult) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4723 (all N_1 all M hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,N_1),M) = hAPP_i1732201573de_int(quickcheck_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,quickcheck_int_of(N_1)),quickcheck_int_of(M)))) # label(fact_4560_plus__code__int__def) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4724 (all N hAPP_n1699378549t_bool(ord_at238088361st_nat(zero_zero_nat),N) = hAPP_n1699378549t_bool(ord_atMost_nat,N)) # label(fact_5185_atLeast0AtMost) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4725 (all R_3 all P_3 all Q_2 (hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2) = normalize(R_3) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Q_2)))) # label(fact_4659_normalize__denom__pos) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4726 (all X_2 all Y_2 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Y_2)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_2)),Y_2))) # label(fact_11_power2__sum) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4727 (all I quickcheck_nat_of(I) = hAPP_int_nat(nat,quickcheck_int_of(I))) # label(fact_4268_nat__of__def) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4728 (all U all Ma all N all I_1 all J (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U)),N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),I_1)),U)),N)))))) # label(fact_1840_nat__less__add__iff2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4729 (all K_2 all L_2 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),L_2)) -> hAPP_int_int(z3mod(K_2),L_2) = hAPP_int_int(div_mod_int(K_2),L_2)) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),L_2)) -> hAPP_int_int(z3mod(K_2),L_2) = hAPP_int_int(div_mod_int(K_2),hAPP_int_int(uminus_uminus_int,L_2))))) # label(fact_3948_z3mod__def) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4730 (all A_3 rcis(zero_zero_real,A_3) = zero_zero_complex) # label(fact_4260_rcis__zero__mod) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4731 (all B_137 all A_161 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_137),zero_zero_real)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_161),zero_zero_real)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_161)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_137)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_161),B_137))))) # label(fact_1180_split__mult__pos__le) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4732 (all J_2 all K_2 all A_3 all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(K_2),zero_zero_int),P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),zero_zero_int),P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),J_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),K_2))),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),K_2),P_3))))))))) # label(fact_2223_MultInv__zcong__prop3) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4733 (all I (is_int(I) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),zero_zero_int)) -> I = hAPP_int_int(abs_abs_int,I)) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),zero_zero_int)) -> hAPP_int_int(uminus_uminus_int,I) = hAPP_int_int(abs_abs_int,I)))) # label(fact_3541_zabs__def) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4734 (all X_1 all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),hAPP_int_int(multInv(P_3),X_1))),one_one_int),P_3)))))) # label(fact_2227_MultInv__prop2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4735 (all B_2 all M all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),A_3)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M))))))) # label(fact_1392_zcong__not) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4736 (all C_75 all A_156 all B_132 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_156),B_132)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_75)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_75),A_156)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_75),B_132)))))) # label(fact_1204_comm__mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4737 (all V_2 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V_2),pls)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(number_number_of_nat,V_2)),one_one_nat) = zero_zero_nat) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V_2),pls)) -> hAPP_int_nat(number_number_of_nat,hAPP_int_int(pred,V_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(number_number_of_nat,V_2)),one_one_nat)))) # label(fact_4037_nat__number__of__diff__1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4738 (all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arccos,Y_1)),pi)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(arccos,Y_1)))))) # label(fact_3614_arccos__bounded) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4739 (all N_55 all A_205 all B_156 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_205),B_156)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_205)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_205),N_55)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_156),N_55)))))) # label(fact_657_power__mono) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4740 (all X_1 all Y_1 hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Y_1)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_1))) # label(fact_4968_transfer__int__nat__gcd_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4741 (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),X_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),one_one_real)) -> hBOOL(hAPP_real_bool(isCont_real_real(arccos),X_2))))) # label(fact_5171_isCont__arccos) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4742 (all A_176 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,zero_z126310315umeral),A_176) = zero_z126310315umeral) # label(fact_1046_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4743 (all X_1 all Y_1 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_1),Y_1) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_complex_real(re,X_1)),hAPP_complex_real(re,Y_1))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_complex_real(im,X_1)),hAPP_complex_real(im,Y_1)))) # label(fact_4133_complex__add__def) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4744 (all A_20 all B_17 (is_int(B_17) -> (zero_zero_int != B_17 -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_20),B_17)),one_one_int) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_20),B_17)),B_17)))) # label(fact_2651_div__add__self2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4745 (all N hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_eq_int),N)))))) # label(fact_4750_bdd__int__set__le__finite) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4746 (all A_213 all N_61 all B_158 (hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_158),N_61) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_213),N_61) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_213)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_158)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_61)) -> B_158 = A_213))))) # label(fact_576_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4747 (all M all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N_1)),N_1)) = hAPP_nat_nat(div_mod_nat(M),N_1)) # label(fact_2747_Divides_Omod__div__equality_H) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4748 (all Z_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_2),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),Z_2)),Z_2)),zero_zero_int)))) # label(fact_755_odd__less__0) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4749 (all A all E_2 all C all B all D (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B)),E_2)),C)),D)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),E_2)),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),E_2)),D))))) # label(fact_1573_le__add__iff1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4750 (all X_2 hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),sin_coeff)),hAPP_r474017924t_real(power_power_real,X_2))),hAPP_real_real(sin,X_2)))) # label(fact_4863_sin__converges) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4751 (all X_2 all Y_2 (zero_zero_real = X_2 & zero_zero_real = Y_2 <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),zero_zero_real)))) # label(fact_849_sum__power2__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4752 (all N_1 all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M)) -> hAPP_nat_nat(div_mod_nat(M),N_1) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)),N_1))) # label(fact_2517_le__mod__geq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4753 (all X_2 all Y_2 (-hBOOL(hAPP_int_bool(even_odd_even_int,X_2)) & -hBOOL(hAPP_int_bool(even_odd_even_int,Y_2)) | hBOOL(hAPP_int_bool(even_odd_even_int,X_2)) & hBOOL(hAPP_int_bool(even_odd_even_int,Y_2)) <-> hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Y_2))))) # label(fact_2893_even__sum) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4754 (all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),one_one_nat)) = N_1)) # label(fact_3182_Suc__diff__1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4755 (all X_2 all X_3 (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),X_3)) -> ((exists Z all X (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X),X_3)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Z)))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),comple124823625p_real(X_3)))))) # label(fact_4638_Sup__upper__EX) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4756 (all Y_1 all X_1 all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(powr,Y_1),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3))))))) # label(fact_4403_powr__less__mono2__neg) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4757 (all W ((-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,W))) -> hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,W)),hAPP_int_nat(number_number_of_nat,W))) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,W))) & (hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,W))) -> zero_zero_nat = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,W))))) # label(fact_4779_nat__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4758 (all X_1 all Y_1 all Xa_2 all Ya hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Ya),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Ya),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_3349_real__sqrt__sum__squares__mult__squared__eq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4759 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(div_mod_nat(M),N_1)),N_1)))) # label(fact_2519_mod__le__divisor) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4760 (all A_3 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),zero_zero_nat))) # label(fact_2277_gcd__lcm__complete__lattice__nat_Otop__greatest) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4761 (all R_3 all S_1 hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(number267125858f_real,R_3)),hAPP_int_real(number267125858f_real,S_1)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,ratreal(hAPP_int_rat(number_number_of_rat,R_3))),ratreal(hAPP_int_rat(number_number_of_rat,S_1)))) # label(fact_4713_Ratreal__number__of__quotient) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4762 (all A all A_1 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma)) -> (hBOOL(hAPP_int_bool(is_RRset(A_1),Ma)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),A_1)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(rRset2norRR(A_1,Ma),A)),hAPP_i1948725293t_bool(norRRset,Ma))))))) # label(fact_4591_RRset2norRR__correct2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4763 (all C all D all A all B all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),D)),Ma)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),D)),Ma))))) # label(fact_1164_zcong__zmult__prop1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4764 (all X_1 all Y_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1)))) # label(fact_4158_termination__basic__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4765 (all B_92 all A_113 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_113)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_92)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_113),B_92)))))) # label(fact_1889_add__pos__nonneg) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4766 (all N all Ma all F ((all N_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N_2)),hAPP_nat_nat(F,hAPP_nat_nat(suc,N_2)))) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,hAPP_nat_nat(suc,N_2))),hAPP_nat_nat(F,N_2))))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),Ma)) <-> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,Ma)),hAPP_nat_nat(F,N))) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N)),hAPP_nat_nat(F,Ma)))))) # label(fact_3263_dvd_Olift__Suc__mono__less__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4767 (all M all N_1 (is_int(M) -> (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N_1) -> hAPP_int_int(number_number_of_int,min) = M | M = one_one_int))) # label(fact_967_pos__zmult__eq__1__iff__lemma) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4768 (all X_36 all Y_28 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_36),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_28),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),Y_28)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,X_36),Y_28))))) # label(fact_842_power2__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4769 (all A_195 all N_47 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_195),N_47)),A_195) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_195),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_195),N_47))) # label(fact_808_power__commutes) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4770 (all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arcsin,Y_1))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arcsin,Y_1)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) & Y_1 = hAPP_real_real(sin,hAPP_real_real(arcsin,Y_1))))) # label(fact_3743_arcsin) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4771 (all M all K_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(fact_fact_int,M)),hAPP_int_int(fact_fact_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M),K_2)))))) # label(fact_4282_fact__mono__int__aux) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4772 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),pi)) -> X_1 = hAPP_real_real(arccos,hAPP_real_real(cos,X_1))))) # label(fact_3602_arccos__cos) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4773 (all A_3 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(div_mod_int(A_3),B_2)),B_2)))) # label(fact_2435_pos__mod__bound) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4774 (all V (zero_zero_nat = hAPP_int_nat(number_number_of_nat,V) <-> hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))))) # label(fact_4049_neg__number__of__pred__iff__0) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4775 (all C_59 all A_125 all B_104 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_59),A_125)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_59),B_104))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_59)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_125),B_104))))) # label(fact_1449_mult__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4776 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N_1))))) # label(fact_2820_Suc__mono) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4777 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(fact_fact_nat,M)),hAPP_nat_nat(fact_fact_nat,N_1))))) # label(fact_3801_fact__mono__nat) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4778 (all A_102 all C_45 all B_85 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_102),C_45)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_85),C_45))) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_102),B_85)))) # label(fact_2004_add__le__imp__le__right) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4779 (all R_3 all A_3 hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_3)),hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(cos,A_3)),hAPP_real_real(sin,A_3)))) = hAPP_real_real(abs_abs_real,R_3)) # label(fact_3997_cmod__complex__polar) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4780 (all R_2 hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(plus_plus_real,R_2)),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,uminus_uminus_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),suc)))),R_2,sequentially))) # label(fact_5045_LIMSEQ__inverse__real__of__nat__add__minus) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4781 (all A_198 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_198),one_one_nat) = A_198) # label(fact_748_power__one__right) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4782 (all F all G all N hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,F),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,G),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))) = hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f335634176l_real(cOMBS_337378985l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),F)),G)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N)))) # label(fact_4909_sum__split__even__odd) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4783 (all X_2 all A (hBOOL(monoseq_real(A)) -> (hBOOL(tendsto_nat_real(A,X_2,sequentially)) -> (all M_1 all N_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A,M_1)),hAPP_nat_real(A,N_2))))) & (all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A,N_2)),X_2))) | (all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),hAPP_nat_real(A,N_2)))) & (all M_1 all N_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A,N_2)),hAPP_nat_real(A,M_1)))))))) # label(fact_5036_monoseq__le) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4784 (all N_1 all V_2 ((-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(succ,V_2))),N_1) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),N_1))) & (hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N_1) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),N_1))))) # label(fact_3291_Suc__nat__number__of__add) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4785 (all A_170 all B_145 all K_7 (hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_145),K_7) = A_170 -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,B_145),A_170)))) # label(fact_1103_dvdI) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4786 (all M all N_1 all R_5 all S_3 all S_1 all T_2 all T_1 all R_3 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_3),zero_zero_int)) -> hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,R_5),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,S_3),T_2)) = xzgcda(M,N_1,R_5,R_3,S_3,S_1,T_2,T_1)) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_3),zero_zero_int)) -> xzgcda(M,N_1,R_5,R_3,S_3,S_1,T_2,T_1) = xzgcda(M,N_1,R_3,hAPP_int_int(div_mod_int(R_5),R_3),S_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,S_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,R_5),R_3)),S_1)),T_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,T_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,R_5),R_3)),T_1)))))) # label(fact_4199_xzgcda_Osimps) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4787 (all A_3 all B_2 all C_1 hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)),C_1) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),C_1))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(div_mod_int(B_2),C_1))),C_1))) # label(fact_2638_zdiv__zmult1__eq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4788 (all X_2 all Y_2 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(number_number_of_rat,X_2)),hAPP_int_rat(number_number_of_rat,Y_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Y_2)))) # label(fact_92_less__number__of) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4789 (all R_3 exists X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_P1175774780nt_int(product_snd_int_int,X))) & hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_P1175774780nt_int(product_fst_int_int,X)),hAPP_P1175774780nt_int(product_snd_int_int,X)) = one_one_int & (all Y (R_3 = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_P1175774780nt_int(product_fst_int_int,Y)),hAPP_P1175774780nt_int(product_snd_int_int,Y)) & hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_P1175774780nt_int(product_fst_int_int,Y)),hAPP_P1175774780nt_int(product_snd_int_int,Y)) = one_one_int & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_P1175774780nt_int(product_snd_int_int,Y))) -> Y = X)) & R_3 = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_P1175774780nt_int(product_fst_int_int,X)),hAPP_P1175774780nt_int(product_snd_int_int,X)))) # label(fact_5023_quotient__of__unique) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4790 (all K_1 all Ma all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K_1)),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K_1)),N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)))) # label(fact_3071_Suc__mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4791 (all N_13 all A_7 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_7)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_7),hAPP_nat_nat(suc,N_13)))))) # label(fact_3119_power__gt1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4792 (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),one_one_real)) -> hBOOL(bseq_real(hAPP_r474017924t_real(power_power_real,X_2)))))) # label(fact_4749_Bseq__realpow) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4793 (all A_70 all B_62 all C_30 hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_70),B_62)),C_30) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_70),C_30)),hAPP_C498520661umeral(div_mo1740067990umeral(B_62),C_30))),C_30)) # label(fact_2324_mod__mult__eq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4794 (all Y_2 all X_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),X_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_2)) -> (zero_zero_nat = X_2 & Y_2 = zero_zero_nat <-> zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_2),Y_2))))) # label(fact_2168_add__nonneg__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4795 (all N all X_2 (X_2 != one_one_real -> hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),N)),one_one_real)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_2),one_one_real)) = hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_r474017924t_real(power_power_real,X_2)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N)))) # label(fact_4902_sumr__geometric) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4796 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N_1) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)),N_1))))) # label(fact_3214_le__div__geq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4797 (all N_1 all M zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),M))) # label(fact_1660_diff__add__0) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4798 (all Z_2 all Ma ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_2),hAPP_nat_int(semiri1621563631at_int,Ma)))) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_2)) -> zero_zero_nat = Ma) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_int_nat(nat,Z_2)),Ma)))) # label(fact_3224_nat__dvd__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4799 (all Code_numeral_3 code_S1047413653umeral(Code_numeral_3) != zero_z126310315umeral) # label(fact_4080_code__numeral_Osimps_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4800 (all Lx_2 all Ly all Rx_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_2),Ly)),Rx_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ly),Rx_2))) # label(fact_166_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4801 (all Ma all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(semiri984289939at_nat,Ma)),hAPP_nat_nat(semiri984289939at_nat,N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)))) # label(fact_2951_of__nat__le__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4802 (all X_24 all Y_18 all A_120 all B_100 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_24),Y_18)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_120),B_100)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_24),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Y_18),B_100))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_24),A_120)),B_100))) # label(fact_1539_mult__diff__mult) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4803 (all P all Q all R hAPP_r1195171167l_real(hAPP_f617735571l_real(hAPP_f1730463541l_real(cOMBS_511527878l_real,P),Q),R) = hAPP_r1195171167l_real(hAPP_r2019696015l_real(P,R),hAPP_real_real(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__prod_Itc) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4804 (all C_81 all A_167 all B_142 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_167),B_142)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_167),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_142),C_81))))) # label(fact_1124_dvd__mult2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4805 (all X_1 hAPP_real_real(sin,X_1) = hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X_1))) # label(fact_3733_sin__cos__eq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4806 (all A_91 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_91),one_on1684967323de_int) = A_91) # label(fact_2105_mult_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4807 (all C_70 all A_145 all B_122 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_145),B_122)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_70)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_70),A_145)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_70),B_122)))))) # label(fact_1260_comm__mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4808 (all A_3 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_21_cube__square) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4809 (all A_195 all N_47 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_195),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_195),N_47)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_195),N_47)),A_195)) # label(fact_807_power__commutes) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4810 (all X_30 all N_34 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_34)) | X_30 = one_on1684967323de_int -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,X_30),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_30),N_34))))) # label(fact_995_dvd__power) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4811 (all Z_1 all W (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_1),W)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),Z_1)) -> Z_1 = W))) # label(fact_1520_real__le__antisym) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4812 (all Y_1 (Y_1 != zero_z126310315umeral -> -(all Code_numeral Y_1 != code_S1047413653umeral(Code_numeral)))) # label(fact_4090_code__numeral_Oexhaust) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4813 (all W_1 all Y_2 all X_2 all Z_2 (Y_2 = Z_2 | W_1 = X_2 <-> hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,W_1),Y_2)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_2),Z_2)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,W_1),Z_2)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_2),Y_2)))) # label(fact_299_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4814 (all A_163 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,one_one_nat),A_163))) # label(fact_1154_one__dvd) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4815 (all X_1 all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),Y_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sin,Y_1)),hAPP_real_real(sin,X_1))))))) # label(fact_3731_sin__monotone__2pi_H) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4816 (all N_1 hAPP_nat_nat(size_size_nat,N_1) = N_1) # label(fact_4058_nat__size) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4817 (all N_58 ((N_58 = zero_zero_nat -> one_one_rat = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,zero_zero_rat),N_58)) & (N_58 != zero_zero_nat -> zero_zero_rat = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,zero_zero_rat),N_58)))) # label(fact_600_power__0__left) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4818 (all X_1 all M all Y_1 hAPP_int_int(div_mod_int(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),Y_1)),M) = hAPP_int_int(div_mod_int(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(div_mod_int(X_1),M)),Y_1)),M)) # label(fact_2405_zpower__zmod) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4819 (all A_179 all B_151 (zero_z126310315umeral = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_179),B_151) -> A_179 = zero_z126310315umeral | zero_z126310315umeral = B_151)) # label(fact_1021_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4820 (all X_2 all P_5 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),hAPP_i1948725293t_bool(sRStar,P_5))) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_2),zero_zero_int),P_5)))) # label(fact_2522_StandardRes__SRStar__prop1a) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4821 (all C_8 all A_24 all B_21 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_24),B_21)) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_21),A_24)),C_8) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_21),C_8)),A_24))) # label(fact_2603_dvd__div__mult) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4822 (all A_128 all C_62 all B_107 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_128),C_62)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_107),C_62))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_62)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_128),B_107))))) # label(fact_1431_mult__right__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4823 (all M all N_1 all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (N_1 != zero_zero_nat -> (N_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_3),M) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)))))) # label(fact_4182_prime__factor__lt) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4824 (all X_2 hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),X_2))))) # label(fact_4762_bdd__nat__set__l__finite) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4825 (all Ma all N all A (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A)) -> (hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A),Ma) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A),N) <-> Ma = N))) # label(fact_876_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4826 (all A_198 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_198),one_one_nat) = A_198) # label(fact_746_power__one__right) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4827 (all A_165 all B_140 hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_165),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_165),B_140)))) # label(fact_1134_dvd__triv__left) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4828 (all Lx all Rx all Ry hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx),Ry)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx),Ry))) # label(fact_180_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4829 (all X_2 all Y_2 (hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_2),Y_2))) <-> hBOOL(hAPP_int_bool(even_odd_even_int,X_2)) | hBOOL(hAPP_int_bool(even_odd_even_int,Y_2)))) # label(fact_2888_Parity_Oeven__product) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4830 (all M all K_2 all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)) # label(fact_1644_diff__cancel2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4831 (all N_1 all X_1 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(exp_real,X_1)),N_1) = hAPP_real_real(exp_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),X_1))) # label(fact_3845_exp__real__of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4832 (all A_207 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_207),zero_z891286103de_int) = A_207) # label(fact_634_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4833 (all B_94 all C_52 all A_115 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_115)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_94),C_52)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_94),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_115),C_52)))))) # label(fact_1876_add__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4834 (all Lx_5 all Ly_3 all Rx_5 all Ry_3 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_5),Ly_3)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_5),Ry_3)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_5),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_5),Ly_3)),Ry_3))) # label(fact_148_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4835 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(uminus_uminus_real,pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sin,X_1)),zero_zero_real))))) # label(fact_3735_sin__less__zero) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4836 (all P all Q all R hAPP_bool_bool(P,hAPP_P603027463t_bool(Q,R)) = hAPP_P603027463t_bool(hAPP_f237669397t_bool(hAPP_f513299955t_bool(cOMBB_1344650379nt_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__prod_Itc__Int__O) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4837 (all C_1 all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_3),B_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_1),A_3)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,A_3),C_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,B_2),C_1)))))) # label(fact_1666_diff__less__mono) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4838 (all A all B (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A),B) <-> (all D_2 (is_int(D_2) -> (D_2 = one_one_int <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_2),A)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_2),B)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),D_2))))))) # label(fact_5018_coprime__int) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4839 (all Y_2 all X_2 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_2),Y_2)),zEven)))) # label(fact_2898_even__times__either) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4840 (all N_1 hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,N_1)) = archim1246769320r_real(hAPP_real_real(uminus_uminus_real,hAPP_nat_real(real_nat,N_1)))) # label(fact_4032_floor__minus__real__of__nat) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4841 (all M all N_1 all K_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),K_2))) # label(fact_1785_add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4842 (all X_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),hAPP_int_int(bit1,pls))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(number_number_of_rat,X_2)),one_one_rat)))) # label(fact_292_le__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4843 (all C all A all B (zero_zero_rat = C | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A),B)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),A)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),B))))) # label(fact_1376_dvd__mult__cancel__left) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4844 (all F all A_1 hBOOL(hAPP_f448129468l_bool(nat_tr876908586nt_nat(F),A_1))) # label(fact_2785_transfer__morphismI) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4845 (all P (is_bool(P) -> P = fFalse | fTrue = P)) # label(help_fTrue_1_1_T) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4846 (all A all P_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> hBOOL(hAPP_f448129468l_bool(finite_finite_int,comple219730294t_bool(setS(A,P_5)))))) # label(fact_3352_Union__SetS__finite) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4847 (all X_2 all A_1 (hBOOL(hAPP_int_bool(A_1,X_2)) <-> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A_1)))) # label(fact_199_mem__def) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4848 (all A all E_2 all C all B all D (hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A),B)),E_2)),C) = D <-> hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A),E_2)),C) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B),E_2)),D))) # label(fact_1553_eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4849 (all X_2 hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,hAPP_int_real(real_int,X_2)),ring_1_Ints_real))) # label(fact_4759_Ints__real__of__int) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4850 (all P all Q P = hAPP_f957591787ol_nat(cOMBK_1643584600t_bool(P),Q)) # label(help_COMBK_1_1_COMBK_000tc__Nat__Onat_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4851 (all A_32 all B_28 (zero_zero_nat != B_28 -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_32),B_28)),B_28) = A_32)) # label(fact_2570_div__mult__self2__is__id) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4852 (all N_19 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),hAPP_nat_nat(suc,N_19)) = zero_zero_nat) # label(fact_2998_power__0__Suc) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4853 (all C (hBOOL(vanishes(cOMBK_rat_nat(C))) <-> zero_zero_rat = C)) # label(fact_5131_vanishes__const) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4854 (all V_3 all K_2 all V_2 ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_2),V_3))),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_3)),K_2))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls)) -> zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_3)),K_2))))) # label(fact_1008_nat__number__of__mult__left) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4855 (all C_44 all D_8 all A_101 all B_84 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_101),B_84)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_44),D_8)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_101),C_44)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_84),D_8)))))) # label(fact_2010_add__mono) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4856 (all P_1 all N all K_1 (is_int(K_1) & is_int(N) -> ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),zero_zero_int)) -> (all I_2 all J_1 (is_int(I_2) & is_int(J_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),J_1)) & hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),I_2)),J_1) = N & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J_1),zero_zero_int)) -> hBOOL(hAPP_int_bool(P_1,J_1)))))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_1)) -> (all I_2 all J_1 (is_int(J_1) & is_int(I_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,J_1),K_1)) & hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),I_2)),J_1) = N & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),J_1)) -> hBOOL(hAPP_int_bool(P_1,J_1)))))) & (K_1 = zero_zero_int -> hBOOL(hAPP_int_bool(P_1,N))) <-> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(div_mod_int(N),K_1)))))) # label(fact_2462_split__zmod) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4857 (all L all U hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_gr660468384an_nat(L),U)))) # label(fact_4630_finite__greaterThanLessThan) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4858 (all M_9 all N_23 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,M_9)),hAPP_nat_nat(semiri984289939at_nat,N_23))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_23)))) # label(fact_2947_of__nat__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4859 (all X_1 all N_1 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(abs_abs_int,X_1)),N_1)))) # label(fact_3434_zero__le__zpower__abs) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4860 (all N_1 zero_zero_real = hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),pi))) # label(fact_3687_tan__npi) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4861 (all A_50 all C_15 all B_42 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,C_15),B_42)) -> hAPP_Q1762011733de_int(div_mo231679042de_int(A_50),C_15) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(div_mo231679042de_int(A_50),B_42)),C_15))) # label(fact_2387_mod__mod__cancel) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4862 (all X_2 all Y_2 (is_int(X_2) & is_int(Y_2) -> (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_2),X_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_2),Y_2)) = zero_zero_int <-> zero_zero_int = X_2 & zero_zero_int = Y_2))) # label(fact_690_sum__squares__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4863 (all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1) = hAPP_nat_nat(suc,hAPP_nat_nat(suc,N_1))) # label(fact_3198_add__2__eq__Suc) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4864 (all C_88 all A_180 all B_152 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_180),B_152)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,B_152),C_88)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_180),C_88))))) # label(fact_987_dvd__trans) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4865 (all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,N_1),zero_zero_nat) = zero_zero_nat) # label(fact_5121_min__0R) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4866 (all A_102 all C_45 all B_85 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_102),C_45)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_85),C_45))) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_102),B_85)))) # label(fact_2005_add__le__imp__le__right) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4867 (all X_1 all Y_1 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1)),Y_1))) # label(fact_514_real__sum__squared__expand) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4868 (all B_126 all A_150 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_150)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_126),zero_zero_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_150),B_126)),zero_zero_nat))))) # label(fact_1234_mult__nonneg__nonpos) # label(axiom) # label(non_clause). [assumption]. 5.74/5.76 4869 (all F all A_1 (hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1)) -> ((all X (is_int(X) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(F,X)))))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),big_co1548731110nt_int(F,A_1)))))) # label(fact_4815_transfer__nat__int__sum__prod__closure_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4870 (all A_20 all B_17 (zero_zero_nat != B_17 -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_20),B_17)),B_17) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_20),B_17)),one_one_nat))) # label(fact_2652_div__add__self2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4871 (all N_48 all A_199 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_199)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_199),one_on1645066479umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_199),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_199),N_48))),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_199),N_48)))))) # label(fact_723_power__Suc__less) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4872 (all B_2 all A_3 (A_3 != zero_zero_nat -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),A_3)))) # label(fact_4345_gcd__le1__nat) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4873 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int)) -> hAPP_P1175774780nt_int(product_snd_int_int,hAPP_P1975530577nt_int(negateSnd,negDivAlg(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2)))) = hAPP_int_int(div_mod_int(A_3),B_2)))) # label(fact_4236_mod__pos__neg) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4874 (all W_17 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(number_number_of_nat,W_17)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,W_17)),hAPP_int_nat(number_number_of_nat,W_17))) # label(fact_19_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4875 (all N_1 zero_zero_real = hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,pi),hAPP_nat_real(real_nat,N_1)))) # label(fact_3639_sin__npi2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4876 (all Lx_1 all Rx_1 all Ry_1 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_1),Rx_1)),Ry_1) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_1),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Rx_1),Ry_1))) # label(fact_171_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4877 (all M all N_1 hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,M)),hAPP_nat_real(real_nat,N_1))) # label(fact_3650_real__of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4878 (all A_174 all B_148 all C_86 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_174),B_148)),C_86) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_174),C_86)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_148),C_86))) # label(fact_1074_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4879 (all P_1 all A0 all A1 (hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(upto_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A0),A1))) -> ((all I_2 all J_1 (is_int(I_2) & is_int(J_1) -> (hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(upto_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,I_2),J_1))) -> ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),J_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I_2),one_one_int)),J_1))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,I_2),J_1)))))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A0),A1))))) # label(fact_4572_upto_Opinduct) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4880 (all X_48 all N_69 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_48),N_69)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_48),N_69)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_48),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_69))) # label(fact_46_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4881 (all A_203 all N_53 all B_155 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_203),N_53)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_155),N_53))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_155)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_203),B_155))))) # label(fact_669_power__less__imp__less__base) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4882 (all A_189 all N_40 all N_39 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_40),N_39)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,one_one_rat),A_189)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_189),N_40)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_189),N_39)))))) # label(fact_871_power__increasing) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4883 (all X_2 (is_int(X_2) -> (X_2 = zero_zero_int <-> hAPP_int_real(real_int,X_2) = zero_zero_real))) # label(fact_4385_real__of__int__zero__cancel) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4884 (all I all M all N_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N_1))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),N_1)))) # label(fact_4982_dvd__gcd__D2__int) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4885 (all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),K_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),hAPP_int_int(bit1,K_1))))) # label(fact_942_rel__simps_I26_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4886 (all A_225 all M_28 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_225),one_one_real)),M_28) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_225),M_28)),M_28)) # label(fact_394_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4887 (all X_1 all Y_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) -> -(hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))))) # label(fact_1710_dvd_Oless__imp__not__less) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4888 (all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),K_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),hAPP_int_int(bit0,K_1))))) # label(fact_270_rel__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4889 (all Ma hAPP_int_nat(phi,Ma) = hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(norRRset,Ma))) # label(fact_4584_phi__def) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4890 (all B_129 all A_153 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_153),zero_zero_rat)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_129),zero_zero_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_153),B_129)))))) # label(fact_1220_mult__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4891 (all A_183 hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_183),zero_zero_nat) = one_on1645066479umeral) # label(fact_958_power__0) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4892 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(uminus_uminus_real,pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(tan,X_1)),zero_zero_real))))) # label(fact_3620_tan__less__zero) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4893 (all P all Q all R hAPP_P507092031nt_rat(hAPP_f1547007718nt_rat(hAPP_f382317021nt_rat(cOMBB_170828966nt_int,P),Q),R) = hAPP_int_fun_int_rat(P,hAPP_P1175774780nt_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Rat__Orat_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4894 (all A_237 all B_166 all C_97 all D_20 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_237),B_166)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_97),D_20)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_237),C_97)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_166),D_20))) # label(fact_192_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4895 (all Lx_2 all Ly all Rx_2 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_2),Ly)),Rx_2) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_2),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Ly),Rx_2))) # label(fact_168_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4896 (all Q_2 ((all A_2 all B_4 (is_int(B_4) & is_int(A_2) -> (hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_4) = Q_2 -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_4)) -> (zero_zero_int != A_2 -> one_one_int != hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_2),B_4)))))) -> zero_zero_rat = Q_2)) # label(fact_5024_Rat__cases__nonzero) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4897 (all A_36 zero_z891286103de_int = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_36),zero_z891286103de_int)) # label(fact_2540_div__by__0) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4898 (all M all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),M))))) # label(fact_1779_trans__le__add1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4899 (all B_1_1 all B_2_1 is_bool(hAPP_P1463840893t_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__prod_Itc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_Mtc__prod_Itc__) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4900 (all A all C all D all B (is_int(B) & is_int(D) -> (zero_zero_int != B -> (D != zero_zero_int -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B)),hAPP_int_rat(hAPP_int_fun_int_rat(fract,C),D))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),D)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),D))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),D))))))))) # label(fact_4712_less__rat) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4901 (all P all Q all R hAPP_P603027463t_bool(hAPP_f892584630t_bool(hAPP_f1325804733t_bool(cOMBS_480885903t_bool,P),Q),R) = hAPP_int_bool(hAPP_P252494982t_bool(P,R),hAPP_P1175774780nt_int(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oin_004) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4902 (all C_89 all D_17 all A_200 all B_154 all R_8 (zero_zero_rat != R_8 -> (D_17 != C_89 & B_154 = A_200 -> hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_200),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,R_8),C_89)) != hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_154),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,R_8),D_17))))) # label(fact_697_add__scale__eq__noteq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4903 (all X_2 (zero_zero_nat = X_2 <-> zero_zero_nat = X_2)) # label(fact_1897_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4904 (all V all W_1 (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,W_1)),hAPP_int_real(number267125858f_real,V))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,V)),hAPP_int_real(number267125858f_real,W_1))))) # label(fact_87_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4905 (all D_3 all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),B_2)))) # label(fact_4011_coprime__lmul2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4906 (all A_3 all C_1 all B_2 all D_3 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_3),C_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_2),D_3)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,C_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,D_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))))) # label(fact_3344_real__sqrt__sum__squares__triangle__ineq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4907 (all N_63 all M_23 all X_42 all Y_34 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,X_42),Y_34)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_63),M_23)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_42),N_63)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_34),M_23)))))) # label(fact_562_dvd__power__le) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4908 (all K_2 (is_int(K_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2)) -> -(all N_2 hAPP_nat_int(semiri1621563631at_int,N_2) != K_2)))) # label(fact_3258_nonneg__int__cases) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4909 (all C_65 all D_12 all A_131 all B_110 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_131),B_110)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,C_65),D_12)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_131)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_65)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_131),C_65)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_110),D_12)))))))) # label(fact_1416_mult__strict__mono_H) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4910 (all N_1 exists P_4 (hBOOL(hAPP_nat_bool(prime,P_4)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),P_4)))) # label(fact_4221_euclid) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4911 (all K_2 zero_zero_rat = frct(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,K_2),zero_zero_int))) # label(fact_4642_Frct__code__post_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4912 (all N hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,hAPP_nat_real(real_nat,N)),field_1210416355s_real))) # label(fact_4631_Rats__real__nat) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4913 (all A_193 all M_21 all N_45 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_193),M_21)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_193),N_45)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_193),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_21),N_45))) # label(fact_821_power__add) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4914 (all Z_10 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Z_10),Z_10) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Z_10),hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_500_semiring__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4915 (all A_207 A_207 = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_207),zero_zero_complex)) # label(fact_633_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4916 (all X_2 all A (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),hAPP_int_real(real_int,A))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim856651990g_real(X_2)),A)))) # label(fact_4423_ceiling__le__eq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4917 (all B all A all C (hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B),A) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C),A) <-> B = C)) # label(fact_1935_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4918 (all K_1 all I_1 all J (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J)) -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),I_1) = J <-> K_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),I_1)))) # label(fact_1672_le__imp__diff__is__add) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4919 (all X_2 all Y_2 (Y_2 != zero_zero_rat | zero_zero_rat != X_2 <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_2),X_2)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_2),Y_2)))))) # label(fact_740_sum__squares__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4920 (all N_1 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),hAPP_nat_real(real_nat,N_1)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1))) # label(fact_4458_powr__realpow) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4921 (all Y_2 all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_2)) -> (X_2 = Y_2 <-> hAPP_real_real(ln,X_2) = hAPP_real_real(ln,Y_2))))) # label(fact_3427_ln__inj__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4922 (all A_229 all B_162 all C_91 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_229),B_162)),C_91) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_229),C_91)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_162),C_91))) # label(fact_309_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4923 (all N_1 all X_1 (is_int(N_1) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N_1)),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N_1)),one_one_real))) -> N_1 = archim1246769320r_real(X_1))))) # label(fact_4451_floor__eq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4924 (all Q_8 all R_5 all A_3 all B_2 all Q_2 all R_3 (is_int(B_2) & is_int(R_3) & is_int(R_5) -> (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3))) -> (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_8),R_5))) -> (B_2 != zero_zero_int -> R_5 = R_3))))) # label(fact_2789_unique__remainder) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4925 (all K_2 hAPP_int_rat(hAPP_int_fun_int_rat(fract,one_one_int),hAPP_int_int(number_number_of_int,K_2)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,one_one_rat),hAPP_int_rat(number_number_of_rat,K_2))) # label(fact_4666_Fract__1__number__of) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4926 (all A_108 all B_87 all C_47 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_108),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_87),C_47)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_108),B_87)),C_47)) # label(fact_1945_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4927 (all A_31 all B_27 (B_27 != zero_z891286103de_int -> A_31 = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_27),A_31)),B_27))) # label(fact_2576_div__mult__self1__is__id) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4928 (all N_1 all D_3 all A_3 (hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),A_3) = one_one_int -> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N_1)) = one_one_int)) # label(fact_4975_coprime__exp__int) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4929 (all A_3 all B_2 all K_2 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),B_2)),hAPP_int_int(div_mod_int(A_3),B_2))),K_2)) # label(fact_2640_zdiv__zmod__equality2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4930 (all M all N_1 all I (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),I)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,I),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,I),N_1))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))))) # label(fact_922_nat__power__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4931 (all N all Ma (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N)),one_one_real)),hAPP_int_real(real_int,Ma))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,N),Ma)))) # label(fact_4445_int__less__real__le) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4932 (all Lx_5 all Ly_3 all Rx_5 all Ry_3 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_5),Ly_3)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_5),Ry_3)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_5),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_5),Ly_3)),Ry_3))) # label(fact_149_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4933 (all N_11 all A_6 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_6)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_6),one_on1645066479umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_6),hAPP_nat_nat(suc,N_11))),one_on1645066479umeral))))) # label(fact_3164_power__Suc__less__one) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4934 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> (hAPP_real_real(sin,X_1) = zero_zero_real -> (exists N_2 (X_1 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) & hBOOL(hAPP_nat_bool(even_odd_even_nat,N_2))))))) # label(fact_3792_sin__zero__lemma) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4935 (all A_192 all M_20 all N_43 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_192),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_20),N_43)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_192),M_20)),N_43)) # label(fact_834_power__mult) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4936 (all A_198 (is_int(A_198) -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_198),one_one_nat) = A_198)) # label(fact_749_power__one__right) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4937 (all X_17 all Y_13 all Ya_2 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_17),Y_13)),Ya_2) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_17),Ya_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_13),Ya_2))) # label(fact_2484_mult__left_Oadd) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4938 (all A_173 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,zero_zero_complex),A_173)) -> A_173 = zero_zero_complex)) # label(fact_1080_dvd__0__left) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4939 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A))))) # label(fact_4716_zero__less__Fract__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4940 (all P all Q all R hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f894608603t_bool(cOMBB_bool_bool_nat,P),Q),R) = hAPP_bool_bool(P,hAPP_nat_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4941 (all D_3 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(abs_abs_int,D_3)),D_3))) # label(fact_3418_zdvd__self__abs2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4942 (all B_2 all A_3 (is_int(A_3) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),B_2)) -> A_3 = hAPP_int_int(div_mod_int(A_3),B_2))))) # label(fact_2451_mod__pos__pos__trivial) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4943 (all N_48 all A_199 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_199)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_199),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_199),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_199),N_48))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_199),N_48)))))) # label(fact_722_power__Suc__less) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4944 (all Y_1 all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(div_mod_int(X_1),Y_1)))))) # label(fact_2438_Divides_Otransfer__nat__int__function__closures_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4945 (all R_3 all A_3 hAPP_complex_real(im,rcis(R_3,A_3)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_real_real(sin,A_3))) # label(fact_4262_Im__rcis) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4946 (all N_60 all X_41 all Y_33 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,X_41),Y_33)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_41),N_60)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_33),N_60))))) # label(fact_584_dvd__power__same) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4947 (all Z1 all Z2 all Z3 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z2),Z3)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z1),Z2)),Z3)) # label(fact_1597_real__mult__assoc) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4948 (all P_1 all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) -> (all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_2))),hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),hAPP_nat_nat(suc,N_2)))))))) # label(fact_4244_Bolzano__bisect__fst__le__Suc) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4949 (all Z_1 all X_1 all Y_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y_1),Z_1))))) # label(fact_4159_termination__basic__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4950 (all A_3 all B_2 all C_1 all D_3 all N_1 (hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,D_3),N_1) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)),C_1) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N_1)) -> (one_one_int = hAPP_int_int(legacy_zgcd(A_3),B_2) -> (hAPP_int_int(legacy_zgcd(B_2),C_1) = one_one_int -> (one_one_int = hAPP_int_int(legacy_zgcd(C_1),A_3) -> (exists K exists L_1 exists M_1 (is_int(L_1) & hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,L_1),N_1) = hAPP_int_int(abs_abs_int,B_2) & hAPP_int_int(abs_abs_int,C_1) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,M_1),N_1) & hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K),N_1) = hAPP_int_int(abs_abs_int,A_3) & is_int(M_1) & is_int(K))))))))) # label(fact_4537_int__triple__relprime__power__divisors) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4951 (all X_1 hAPP_real_real(abs_abs_real,hAPP_int_real(real_int,X_1)) = hAPP_int_real(real_int,hAPP_int_int(abs_abs_int,X_1))) # label(fact_4399_real__of__int__abs) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4952 (all A_183 one_on1684967323de_int = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_183),zero_zero_nat)) # label(fact_960_power__0) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4953 (all A_3 all B_2 all C_1 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),C_1)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),C_1))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(div_mod_int(A_3),C_1)),hAPP_int_int(div_mod_int(B_2),C_1))),C_1)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)),C_1)) # label(fact_2564_zdiv__zadd1__eq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4954 (all A_222 A_222 = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_222),hAPP_int_real(number267125858f_real,pls))) # label(fact_418_add__numeral__0__right) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4955 (all C_37 all A_94 all B_77 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_94),B_77)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_94),C_37)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_77),C_37))))) # label(fact_2064_add__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4956 (all P all Q all R hAPP_r1938957037l_bool(hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,P),Q),R) = hAPP_b589554111l_bool(P,hAPP_real_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_007) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4957 (all X_2 all Y_2 (hBOOL(hAPP_P1555980039t_bool(accp_P490777396at_nat(nat_gcd_rel),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,X_2),Y_2))) -> (Y_2 != zero_zero_nat -> hAPP_nat_nat(nat_gcd(X_2),Y_2) = hAPP_nat_nat(nat_gcd(Y_2),hAPP_nat_nat(div_mod_nat(X_2),Y_2))) & (Y_2 = zero_zero_nat -> hAPP_nat_nat(nat_gcd(X_2),Y_2) = X_2))) # label(fact_4621_nat__gcd_Opsimps) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4958 (all X_2 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Y_2)) & X_2 != Y_2 <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Y_2)))) # label(fact_1512_real__less__def) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4959 (all W_1 all X_2 (hAPP_i1732201573de_int(number1226105091de_int,W_1) = X_2 <-> hAPP_i1732201573de_int(number1226105091de_int,W_1) = X_2)) # label(fact_256_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4960 (all X_9 all Q_7 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_9),Q_7)),X_9) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_9),hAPP_nat_nat(suc,Q_7))) # label(fact_3007_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4961 (all A_76 all B_65 hAPP_nat_nat(div_mod_nat(A_76),B_65) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(div_mod_nat(A_76),B_65)),B_65)) # label(fact_2299_mod__mod__trivial) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4962 (all A all B (zero_zero_complex = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A),B) <-> B = A)) # label(fact_2088_eq__iff__diff__eq__0) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4963 (all X_32 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_32),zero_zero_nat) = one_one_rat) # label(fact_954_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4964 (all A_26 all C_9 all B_23 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,C_9),B_23)) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_26),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_23),C_9)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_26),B_23)),C_9))) # label(fact_2595_div__mult__swap) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4965 (all M_12 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,M_12)))) # label(fact_2917_zero__le__imp__of__nat) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4966 (all A_183 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_183),zero_zero_nat) = one_one_nat) # label(fact_956_power__0) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4967 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M)),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)))) # label(fact_2871_Suc__leD) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4968 (all A_3 all B_2 hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(uminus_uminus_int,A_3)),hAPP_int_int(uminus_uminus_int,B_2))) # label(fact_4684_minus__rat__cancel) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4969 (all A_14 all B_11 all C_4 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_14),C_4) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_14),B_11)),B_11)),hAPP_Q1762011733de_int(div_mo231679042de_int(A_14),B_11))),C_4)) # label(fact_2686_div__mod__equality) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4970 (all C_29 all A_69 all B_61 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_29),hAPP_int_int(div_mod_int(A_69),B_61)) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_29),A_69)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_29),B_61))) # label(fact_2326_mod__mult__mult1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4971 (all M all K_2 all N_1 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,K_2),N_1) = one_one_nat -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),M))))) # label(fact_4368_coprime__dvd__mult__nat) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4972 (all X_8 all Q_6 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_8),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_8),Q_6)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_8),hAPP_nat_nat(suc,Q_6))) # label(fact_3016_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4973 (all N_1 (N_1 != zero_zero_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),N_1) = hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),N_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_3232_lemma__Suc__Suc__4n__diff__2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4974 (all A_107 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_107),zero_zero_real) = A_107) # label(fact_1966_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4975 (all X_1 all N_1 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(real_int,X_1)),N_1) = hAPP_int_real(real_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),N_1))) # label(fact_4389_power__real__of__int) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4976 (all A_112 all B_91 all C_51 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_112),B_91)),C_51) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_112),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_91),C_51))) # label(fact_1907_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4977 (all A_233 all C_93 all D_18 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_233),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_93),D_18)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_93),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_233),D_18))) # label(fact_234_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4978 (all X_34 all Y_26 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_34),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_26),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),zero_zero_int))) # label(fact_851_not__sum__power2__lt__zero) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4979 (all P_1 all L ((exists X (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,L),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X),zero_z891286103de_int))) & hBOOL(hAPP_Q2096512830t_bool(P_1,X)))) <-> (exists X hBOOL(hAPP_Q2096512830t_bool(P_1,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,L),X)))))) # label(fact_1809_unity__coeff__ex) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4980 (all Z_4 all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_4)) -> hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),Z_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(nat,Z_1)),hAPP_int_nat(nat,Z_4))))) # label(fact_2757_nat__add__distrib) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4981 (all A_87 all B_76 hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_87),B_76)),B_76) = A_87) # label(fact_2135_add__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4982 (all N_19 zero_zero_int = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,zero_zero_int),hAPP_nat_nat(suc,N_19))) # label(fact_2999_power__0__Suc) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4983 (all Ma all N (hBOOL(hAPP_nat_bool(nat_case_bool(fFalse,hAPP_n1699378549t_bool(ord_less_eq_nat,Ma)),N)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,Ma)),N)))) # label(fact_4876_less__eq__nat_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4984 (all A_109 all B_88 all C_48 (hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_109),C_48) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_109),B_88) -> B_88 = C_48)) # label(fact_1930_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4985 (all A all E_2 all C all B all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B),A)),E_2)),D))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),E_2)),C)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),E_2)),D))))) # label(fact_1575_less__add__iff2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4986 (all A_3 all B_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),B_2)) -> hAPP_int_int(div_mod_int(hAPP_int_int(div_mod_int(A_3),B_2)),M) = hAPP_int_int(div_mod_int(A_3),M)))) # label(fact_2455_zmod__zdvd__zmod) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4987 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K1)),hAPP_int_int(bit0,K2))))) # label(fact_267_less__int__code_I15_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4988 (all X_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),hAPP_int_int(bit1,pls))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,X_2)),one_one_int)))) # label(fact_284_less__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4989 (all X_1 all Y_1 hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(nat_lcm(hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,X_1))),hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,Y_1)))) = hAPP_int_int(int_lcm(X_1),Y_1)) # label(fact_4597_Nitpick_Oint__lcm__def) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4990 (all K_1 all P_1 all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D)) -> ((all X (is_int(X) -> (hBOOL(hAPP_int_bool(P_1,X)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),D)))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_1)) -> (all X (hBOOL(hAPP_int_bool(P_1,X)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),D)))))))))) # label(fact_1826_decr__mult__lemma) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4991 (all Z_1 all W hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),hAPP_int_nat(number_number_of_nat,W))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),hAPP_int_nat(number_number_of_nat,W))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,W)))) # label(fact_4766_zpower__number__of__even) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4992 (all N_41 all A_190 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,one_on1645066479umeral),A_190)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,one_on1645066479umeral),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_190),N_41))))) # label(fact_863_one__le__power) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4993 (all A_165 all B_140 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_165),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_165),B_140)))) # label(fact_1135_dvd__triv__left) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4994 (all A_88 (is_int(A_88) -> A_88 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,one_one_int),A_88))) # label(fact_2121_mult__1__left) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4995 (all N_1 all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(fact_fact_nat,N_1)),hAPP_nat_nat(fact_fact_nat,M))))) # label(fact_3808_fact__dvd) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4996 (all B_1_1 is_bool(monoseq_real(B_1_1))) # label(gsy_c_SEQ_Omonoseq_000tc__RealDef__Oreal) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4997 (all Z_2 (is_int(Z_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(abs_abs_int,Z_2)),one_one_int)) <-> Z_2 = zero_zero_int))) # label(fact_3432_zabs__less__one__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4998 (all C_68 all A_143 all B_120 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_143),B_120)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_68)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_143),C_68)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_120),C_68)))))) # label(fact_1272_mult__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 4999 (all A_3 all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),archim1246769320r_real(X_1))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,A_3)),X_1)))) # label(fact_4417_real__le__floor) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5000 (all N_1 all M quickc495462417de_int(N_1,M) = hAPP_Q1256397320de_int(hAPP_Q952608535de_int(produc1318306967de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,N_1),M)),hAPP_Q1762011733de_int(div_mo231679042de_int(N_1),M))) # label(fact_4267_div__mod__code__int__def) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5001 (all B_126 all A_150 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_150)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_126),zero_z126310315umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_150),B_126)),zero_z126310315umeral))))) # label(fact_1236_mult__nonneg__nonpos) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5002 (all F all B all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B),X_2)) -> ((all X (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X),hAPP_r1134773055l_bool(ord_gr788844697n_real(B),X_2))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(F,X))))) -> (hBOOL(hAPP_real_bool(isCont_real_real(F),X_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(F,X_2))))))) # label(fact_5173_LIM__less__bound) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5003 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),one_one_real)) -> hAPP_real_real(sin,hAPP_real_real(arccos,X_1)) = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3631_sin__arccos) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5004 (all A all A_1 all Ma ((hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),A_1)) & hBOOL(hAPP_int_bool(is_RRset(A_1),Ma)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma)) -> hilbert_Eps_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,zcong(A)),Ma))),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),hAPP_i1948725293t_bool(norRRset,Ma)))) = hAPP_int_int(rRset2norRR(A_1,Ma),A)) & (-(hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma)) & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),A_1)) & hBOOL(hAPP_int_bool(is_RRset(A_1),Ma))) -> hAPP_int_int(rRset2norRR(A_1,Ma),A) = zero_zero_int))) # label(fact_4891_RRset2norRR__def) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5005 (all K_2 all L_2 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(bit1,K_2)),L_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),L_2))),L_2)) # label(fact_474_mult__Bit1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5006 (all M all K_2 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K_2)),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)))) # label(fact_1774_add__leD1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5007 (all Ta all B_1 all D all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),D_1)) -> (all X (is_int(X) -> ((all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B_1)) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) != X)))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Ta))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),D_1)),Ta))))))))) # label(fact_5103_bset_I9_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5008 (all A_3 all N_1 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,cis(A_3)),N_1) = cis(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),A_3))) # label(fact_4152_DeMoivre) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5009 (all A all B (zero_zero_rat = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B) <-> B = A)) # label(fact_2089_eq__iff__diff__eq__0) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5010 (all M zero_zero_nat != hAPP_nat_nat(suc,M)) # label(fact_2853_Suc__not__Zero) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5011 (all Lx_3 all Ly_1 all Rx_3 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_3),Ly_1)),Rx_3) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_3),Rx_3)),Ly_1)) # label(fact_157_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5012 (all M_19 all N_38 all A_188 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_188)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_188),M_19)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_188),N_38))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_19),N_38))))) # label(fact_885_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5013 (all L_2 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,pls),hAPP_int_int(bit1,L_2)) = hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,min),L_2))) # label(fact_1587_diff__bin__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5014 (all X_2 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hBOOL(hAPP_real_bool(isCont_real_real(hAPP_n546711566l_real(root,N)),X_2)))) # label(fact_5158_isCont__real__root) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5015 (all B_112 all C_67 all A_134 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_134)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_112),C_67)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_112),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_134),C_67)))))) # label(fact_1336_pos__add__strict) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5016 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N_1))))) # label(fact_2862_less__SucI) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5017 (all A_202 all N_52 all N_51 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_52),N_51)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_202)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_202),one_one_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_202),N_51)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_202),N_52))))))) # label(fact_675_power__decreasing) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5018 (all X_49 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_49),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_49),X_49)) # label(fact_35_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5019 (all X_2 all Y_2 (zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_2),Y_2) <-> zero_zero_nat = Y_2 & zero_zero_nat = X_2)) # label(fact_4340_gcd__lcm__complete__lattice__nat_Oinf__eq__top__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5020 (all B_72 all C_35 all A_83 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_83)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_72),C_35)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_72),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_83),C_35)))))) # label(fact_2161_add__increasing) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5021 (all P all Q all R hAPP_i1216375074nt_int(hAPP_f1375227553nt_int(hAPP_f1311564564nt_int(cOMBS_1640193733nt_int,P),Q),R) = hAPP_P1145851913nt_int(hAPP_i2104874254nt_int(P,R),hAPP_i1524277240nt_int(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint__027) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5022 (all X_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),hAPP_real_real(sin,X_1)))) # label(fact_3659_sin__ge__minus__one) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5023 (all X_1 all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arctan,X_1)),hAPP_real_real(arctan,Y_1))))) # label(fact_3376_arctan__monotone) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5024 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(sin,X_1)))))) # label(fact_3707_sin__gt__zero) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5025 (all V_13 all W_11 all Z_13 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_13),W_11))),Z_13) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_13)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,W_11)),Z_13))) # label(fact_436_add__number__of__left) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5026 (all X_1 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_377_power2__ge__self) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5027 (all A_1 all Ma (hBOOL(hAPP_int_bool(is_RRset(A_1),Ma)) <-> hAPP_f957591787ol_nat(finite_card_int,A_1) = hAPP_int_nat(phi,Ma) & hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,A_1),rsetR(Ma))))) # label(fact_4593_is__RRset__def) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5028 (all X_4 all Y_4 hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_4),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) = hAPP_real_real(cos,hAPP_real_real(arccos,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_4),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))))) # label(fact_3594_cos__arccos__lemma1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5029 (all K_2 hAPP_int_int(bit0,K_2) != min) # label(fact_929_rel__simps_I45_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5030 (all P all Q all R hAPP_f727283836t_bool(P,hAPP_i1823918693l_bool(Q,R)) = hAPP_i1529485324t_bool(hAPP_f1545556668t_bool(hAPP_f85237525t_bool(cOMBB_389152643ol_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__HOL__Obool_Mtc__HOL) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5031 (all A_226 all B_161 all V_18 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_226),B_161)),hAPP_i769753017umeral(number1443263063umeral,V_18)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_226),hAPP_i769753017umeral(number1443263063umeral,V_18))),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_161),hAPP_i769753017umeral(number1443263063umeral,V_18)))) # label(fact_378_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5032 (all B_1_1 all B_2_1 is_bool(hAPP_complex_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__Complex__Ocomplex_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5033 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hAPP_int_int(div_mod_int(A_3),B_2) = hAPP_P1175774780nt_int(product_snd_int_int,negDivAlg(A_3,B_2))))) # label(fact_4233_mod__neg__pos) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5034 (all N_1 (zero_zero_nat != N_1 -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),one_one_nat)),N_1) = one_one_nat)) # label(fact_4365_coprime__minus__one__nat) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5035 (all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),one_one_real)) -> hAPP_real_real(sin,hAPP_real_real(arcsin,Y_1)) = Y_1))) # label(fact_3702_sin__arcsin) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5036 (all P_1 (-hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) -> ((exists X1 hBOOL(hAPP_nat_bool(P_1,X1))) -> (exists N_2 (hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,N_2))) & -hBOOL(hAPP_nat_bool(P_1,N_2))))))) # label(fact_3954_exists__least__lemma) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5037 (all X_2 hBOOL(hAPP_real_bool(isCont_real_real(sin),X_2))) # label(fact_5163_isCont__sin) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5038 (all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),N_1)))) # label(fact_3751_lemma__STAR__cos) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5039 (all N_13 all A_7 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_7)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_7),hAPP_nat_nat(suc,N_13)))))) # label(fact_3120_power__gt1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5040 (all Ma all W_1 (is_int(W_1) -> ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),W_1)) -> W_1 = hAPP_nat_int(semiri1621563631at_int,Ma)) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),W_1)) -> Ma = zero_zero_nat) <-> Ma = hAPP_int_nat(nat,W_1)))) # label(fact_3170_nat__eq__iff2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5041 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A),B)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),zero_zero_int))))) # label(fact_2555_pos__imp__zdiv__neg__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5042 (all C_57 all A_123 all B_102 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_57),A_123)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_57),B_102))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_57)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_123),B_102))))) # label(fact_1461_mult__left__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5043 (all B_74 all A_85 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_85),zero_z126310315umeral)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_74),zero_z126310315umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_85),B_74)),zero_z126310315umeral))))) # label(fact_2148_add__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5044 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)))) # label(fact_1769_less__imp__le__nat) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5045 (all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1)) -> (exists X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),hAPP_real_real(tan,X))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)))))) # label(fact_3628_lemma__tan__total) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5046 (all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,min)),B_2) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_2),one_one_int))) # label(fact_2469_zmod__minus1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5047 (all K_6 all M_16 all N_29 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,K_6),hAPP_Q1762011733de_int(div_mo231679042de_int(M_16),N_29))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,K_6),N_29)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,K_6),M_16))))) # label(fact_2379_dvd__mod__imp__dvd) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5048 (all Ma all K_1 all N (is_int(K_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N)) -> (K_1 = hAPP_int_int(legacy_zgcd(Ma),N) <-> (exists S_2 exists T (is_int(S_2) & is_int(T) & xzgcd(Ma,N) = hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,K_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,S_2),T)))))))) # label(fact_4529_xzgcd__correct) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5049 (all X_1 all Y_1 all M hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),hAPP_int_int(div_mod_int(Y_1),M))),M) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),Y_1)),M)) # label(fact_2398_zdiff__zmod__right) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5050 (all C_80 all A_164 all B_139 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_164),B_139)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_164),C_80)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_164),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_139),C_80)))))) # label(fact_1147_dvd__add) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5051 (all M all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1)) # label(fact_1723_nat__add__commute) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5052 (all M_7 all N_21 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,M_7)),hAPP_nat_int(semiri1621563631at_int,N_21)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_7),N_21))) # label(fact_2968_of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5053 (all B_14 all A_17 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_14),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_17),B_14))),hAPP_Q1762011733de_int(div_mo231679042de_int(A_17),B_14)) = A_17) # label(fact_2674_mod__div__equality2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5054 (all X_6 all Y_5 hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_6),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_5),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_6),Y_5)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_6),Y_5))) # label(fact_3191_realpow__two__diff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5055 (all A_176 zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,zero_zero_nat),A_176)) # label(fact_1044_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5056 (all Y_3 all X_3 (hBOOL(vanishes(X_3)) -> (hBOOL(vanishes(Y_3)) -> hBOOL(vanishes(hAPP_f2062659861at_rat(hAPP_f1131305186at_rat(cOMBS_nat_rat_rat,hAPP_f1830834240at_rat(hAPP_f1073644437at_rat(cOMBB_1926947at_nat,minus_minus_rat),X_3)),Y_3)))))) # label(fact_5129_vanishes__diff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5057 (all B_143 all A_168 all C_82 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_168),C_82)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_168),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_143),C_82))))) # label(fact_1116_dvd__mult) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5058 (all A_197 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_197),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_781_zero__le__power2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5059 (all A_109 all B_88 all C_48 (hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_109),C_48) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_109),B_88) -> C_48 = B_88)) # label(fact_1928_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5060 (all X_15 all Y_11 all Ya_1 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_15),Ya_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_11),Ya_1)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_15),Y_11)),Ya_1)) # label(fact_2492_mult__left_Odiff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5061 (all X_1 all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arctan,X_1)),hAPP_real_real(arctan,Y_1))))) # label(fact_3374_arctan__monotone_H) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5062 (all A_49 all B_41 zero_zero_nat = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_49),B_41)),B_41)) # label(fact_2408_mod__mult__self2__is__0) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5063 (all Y_1 all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),Y_1)))))) # label(fact_2559_Divides_Otransfer__nat__int__function__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5064 (all A_226 all B_161 all V_18 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_226),hAPP_int_nat(number_number_of_nat,V_18))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_161),hAPP_int_nat(number_number_of_nat,V_18))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_226),B_161)),hAPP_int_nat(number_number_of_nat,V_18))) # label(fact_380_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5065 (all I_1 all J all K_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),J)))) # label(fact_1664_less__diff__conv) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5066 (all A_9 all N_15 all B_8 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_9),hAPP_nat_nat(suc,N_15))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_8),hAPP_nat_nat(suc,N_15)))) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_8)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_9),B_8))))) # label(fact_3106_power__le__imp__le__base) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5067 (all Y_1 all X_1 (hBOOL(hAPP_int_bool(even_odd_even_int,X_1)) -> hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_1))))) # label(fact_2887_even__times__anything) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5068 (all I all U_1 all J_2 all K_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J_2)),U_1)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),U_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_2),U_1)),K_2))) # label(fact_1843_left__add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5069 (all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,M),zero_zero_int)) -> hAPP_int_int(fact_fact_int,M) = zero_zero_int)) # label(fact_4274_fact__neg__int) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5070 (all Ma all N (Ma = N <-> hAPP_n522471361de_int(semiri1424489471de_int,N) = hAPP_n522471361de_int(semiri1424489471de_int,Ma))) # label(fact_2830_of__nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5071 (all N_1 all X_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,N_1)),hAPP_nat_real(real_nat,X_1))),hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N_1),X_1)))),one_one_real))) # label(fact_3700_real__of__nat__div3) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5072 (all Z_1 all X_1 all Y_1 all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),Y_1)),one_one_int),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Y_1),Z_1))),one_one_int),P_3)))) # label(fact_666_zcong__zpower__zmult) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5073 (all B_93 all A_114 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_114)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_93)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_114),B_93)))))) # label(fact_1883_add__nonneg__pos) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5074 (all P_1 ((exists X (is_int(X) & hBOOL(hAPP_nat_bool(P_1,hAPP_int_nat(nat,X))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)))) <-> (exists X1 hBOOL(hAPP_nat_bool(P_1,X1))))) # label(fact_2727_ex__nat) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5075 (all N_1 all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),N_1))))) # label(fact_1396_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5076 (all D all Ta (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),Ta)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(uminus_uminus_int,D)),Ta)))) # label(fact_3519_uminus__dvd__conv_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5077 (all A_213 all N_61 all B_158 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_158),N_61) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_213),N_61) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_213)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_158)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_61)) -> A_213 = B_158))))) # label(fact_575_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5078 (all Ma all N (Ma = N | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_nat_nat(suc,N))))) # label(fact_2858_less__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5079 (all N all Ma (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,Ma)),one_one_real))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),Ma)))) # label(fact_3685_nat__le__real__less) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5080 (all Y_1 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1)) -> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_1),hAPP_real_real(ln,X_1)) = hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),Y_1))))) # label(fact_4459_ln__powr) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5081 (all W_7 hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W_7)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),one_one_int)),hAPP_int_int(number_number_of_int,W_7))) # label(fact_475_double__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5082 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> hAPP_real_real(log(hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X_1) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(log(hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(exp_real,one_one_real))),hAPP_real_real(ln,X_1)))) # label(fact_3881_log__base__10__eq2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5083 (all Q all P (-hBOOL(Q) | hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q)) | -hBOOL(P))) # label(help_fconj_1_1_U) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5084 (all A_3 all B_2 all C_1 all D_3 hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_3),C_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_2),D_3))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_3),D_3)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_2),C_1))) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),B_2)),hAPP_real_complex(hAPP_r265291036omplex(complex,C_1),D_3))) # label(fact_3964_complex__mult) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5085 (all X_2 all Y_2 (Y_2 = X_2 <-> quickcheck_int_of(X_2) = quickcheck_int_of(Y_2))) # label(fact_4253_code__int_Oint__of__inject) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5086 (all Lx_6 all Ly_4 all Rx_6 all Ry_4 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_6),Ly_4)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_6),Ry_4)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_6),Rx_6)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Ly_4),Ry_4))) # label(fact_140_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5087 (all I all K_2 all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_2)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_2)),I) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),I)),K_2))) # label(fact_1670_add__diff__assoc2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5088 (all K_2 all A_3 all B_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),A_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),B_2)),M)))) # label(fact_1161_zcong__scalar2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5089 (all Y_1 all X_1 (zero_zero_real != hAPP_real_real(cos,X_1) -> (hAPP_real_real(cos,Y_1) != zero_zero_real -> hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(tan,X_1)),hAPP_real_real(tan,Y_1))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_1)),hAPP_real_real(cos,Y_1)))))) # label(fact_3612_lemma__tan__add1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5090 (all X_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) <-> -hBOOL(hAPP_int_bool(nat_neg,X_2)))) # label(fact_3274_not__neg__eq__ge__0) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5091 (all X_37 all Y_29 (hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_37),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_29),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),X_37)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Y_29)) -> X_37 = Y_29)))) # label(fact_794_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.77 5092 (all A_230 all M_29 all B_163 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_230),B_163)),M_29) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_230),M_29)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_163),M_29))) # label(fact_305_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5093 (all C_42 all A_99 all B_82 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_99),B_82)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_99),C_42)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_82),C_42))))) # label(fact_2019_add__right__mono) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5094 (all A all B all C all D (hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,C),D) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A),B)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C),D))))) # label(fact_2096_diff__eq__diff__less__eq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5095 (all A all W_1 (zero_zero_nat != hAPP_int_nat(number_number_of_nat,W_1) & zero_zero_complex = A <-> zero_zero_complex = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A),hAPP_int_nat(number_number_of_nat,W_1)))) # label(fact_1386_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5096 (all A all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(big_co1548731110nt_int(cOMBI_int,hAPP_i1948725293t_bool(wset(A),P_5))),one_one_int),P_5)))))) # label(fact_4813_wset__zcong__prod__1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5097 (all Z_1 all Z_4 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_4)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_4),Z_1)) -> hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_1),Z_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat,Z_1)),hAPP_int_nat(nat,Z_4))))) # label(fact_2759_nat__diff__distrib) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5098 (exists K_4 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),K_4)) & (all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(cnj,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X)),K_4)))))) # label(fact_4085_cnj_Opos__bounded) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5099 (all C_69 all A_144 all B_121 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_144),B_121)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_69)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_69),A_144)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_69),B_121)))))) # label(fact_1267_mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5100 (all A all B all C all D (hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,C),D) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B) -> (C = D <-> B = A))) # label(fact_1962_diff__eq__diff__eq) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5101 (all N_13 all A_7 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_7)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_7),hAPP_nat_nat(suc,N_13)))))) # label(fact_3118_power__gt1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5102 (all X_1 hAPP_real_real(uminus_uminus_real,hAPP_int_real(real_int,X_1)) = hAPP_int_real(real_int,hAPP_int_int(uminus_uminus_int,X_1))) # label(fact_4398_real__of__int__minus) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5103 (all A_181 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_181),A_181))) # label(fact_981_dvd__refl) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5104 (all A_68 all C_28 all B_60 hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_68),C_28)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_60),C_28)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(div_mod_nat(A_68),B_60)),C_28)) # label(fact_2331_mod__mult__mult2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5105 (all C_1 all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_1),A_3)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_1),B_2))))) # label(fact_1605_divides__mul__l) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5106 (all X_38 all Y_30 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_38),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_30),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_30)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_38),Y_30))))) # label(fact_783_power2__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5107 (all A_229 all B_162 all C_91 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_229),B_162)),C_91) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_229),C_91)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_162),C_91))) # label(fact_310_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5108 (all A_171 all B_146 all C_84 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_171),B_146)),C_84)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_171),C_84)))) # label(fact_1093_dvd__mult__left) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5109 (all P all Q all R hAPP_r2019696015l_real(P,hAPP_real_real(Q,R)) = hAPP_r2019696015l_real(hAPP_f1591894897l_real(hAPP_f2114902373l_real(cOMBB_237479429l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__p) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5110 (all X_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> hAPP_real_real(hAPP_n546711566l_real(root,N_1),hAPP_real_real(inverse_inverse_real,X_1)) = hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1))))) # label(fact_3939_real__root__inverse__lemma) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5111 (all X_1 all Y_1 ((zero_zero_nat != Y_1 -> hAPP_nat_nat(nat_gcd(Y_1),hAPP_nat_nat(div_mod_nat(X_1),Y_1)) = hAPP_nat_nat(nat_gcd(X_1),Y_1)) & (Y_1 = zero_zero_nat -> hAPP_nat_nat(nat_gcd(X_1),Y_1) = X_1))) # label(fact_4616_nat__gcd_Osimps) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5112 (all A all B all C (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A),B)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),A)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),B)))))) # label(fact_1322_mult__less__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5113 (all L all U hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,U),L)) = hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(ord_at641636577an_int(L),U))) # label(fact_4919_card__atLeastLessThan__int) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5114 (all L_2 all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),L_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_2),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_2),M)))))) # label(fact_1642_diff__less__mono2) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5115 (all C_70 all A_145 all B_122 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_145),B_122)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_70)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_70),A_145)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_70),B_122)))))) # label(fact_1265_comm__mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5116 (all A_104 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,zero_z891286103de_int),A_104) = A_104) # label(fact_1992_add__0__left) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5117 (all Ma all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) -> -hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(bnorRset(A),Ma))))) # label(fact_4291_Bnor__mem__zle__swap) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5118 (all Ma all Z_2 (is_int(Z_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_2)) & Ma = hAPP_int_nat(nat,Z_2) <-> Z_2 = hAPP_nat_int(semiri1621563631at_int,Ma)))) # label(fact_3125_int__eq__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5119 (all Code_numeral_1 code_c271388182l_size(code_S1047413653umeral(Code_numeral_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,code_c271388182l_size(Code_numeral_1)),hAPP_nat_nat(suc,zero_zero_nat))) # label(fact_4061_code__numeral_Osize_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5120 (all B_1_1 all B_2_1 (is_int(B_2_1) -> is_bool(hAPP_int_bool(B_1_1,B_2_1)))) # label(gsy_c_hAPP_000tc__Int__Oint_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5121 (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2)) -> hBOOL(hAPP_real_bool(deriv_real(ln,X_2),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),X_2))))) # label(fact_3869_DERIV__ln__divide) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5122 (all C_78 all D_14 all A_159 all B_135 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_159),B_135)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,C_78),D_14)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_159)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_78)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_159),C_78)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_135),D_14)))))))) # label(fact_1191_mult__mono_H) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5123 (all Lx_1 all Rx_1 all Ry_1 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_1),Rx_1)),Ry_1) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_1),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_1),Ry_1))) # label(fact_175_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5124 (all C all A all B (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C),A)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C),B))) <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A),B)))) # label(fact_2028_add__le__cancel__left) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5125 (all Y_2 all X_2 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),X_2)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Y_2)) -> (hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_2),Y_2) = zero_zero_rat <-> zero_zero_rat = Y_2 & X_2 = zero_zero_rat)))) # label(fact_2171_add__nonneg__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5126 (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,X_2)),one_one_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),one_one_real)))) # label(fact_3333_real__sqrt__lt__1__iff) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5127 (all A_89 (is_int(A_89) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,one_one_int),A_89) = A_89)) # label(fact_2114_mult__1) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5128 (all C_33 all D_5 all A_78 all B_67 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_78),B_67)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_33),D_5)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_78),C_33)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_67),D_5)))))) # label(fact_2211_add__less__le__mono) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5129 (all M_27 all A_224 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_224),one_on1645066479umeral)),M_27) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,M_27),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_224),M_27))) # label(fact_402_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5130 (all A (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),zero_zero_int)))) # label(fact_2196_double__add__less__zero__iff__single__add__less__zero) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5131 (all P_1 all N all K_1 (hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(div_mod_nat(N),K_1))) <-> (K_1 = zero_zero_nat -> hBOOL(hAPP_nat_bool(P_1,N))) & (zero_zero_nat != K_1 -> (all I_2 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),K_1)) -> (N = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),I_2)),J_1) -> hBOOL(hAPP_nat_bool(P_1,J_1)))))))) # label(fact_2524_split__mod) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5132 (all B_92 all A_113 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_113)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_92)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_113),B_92)))))) # label(fact_1887_add__pos__nonneg) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5133 (all C_60 all A_126 all B_105 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_60),A_126)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_60),B_105))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_60)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_126),B_105))))) # label(fact_1445_mult__left__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 5.74/5.78 5134 -(exists X exists Y hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(conj_0) # label(negated_conjecture) # label(non_clause). [assumption]. 12.40/12.39 12.40/12.39 ============================== end of process non-clausal formulas === 12.40/12.39 12.40/12.39 ============================== PROCESS INITIAL CLAUSES =============== 12.40/12.39 12.40/12.39 ============================== PREDICATE ELIMINATION ================= 12.40/12.39 12.40/12.39 ============================== end predicate elimination ============= 12.40/12.39 12.40/12.39 Auto_denials: (non-Horn, no changes). 12.40/12.39 12.40/12.39 Term ordering decisions: 12.40/12.39 Function symbol KB weights: ord_less_int=1. zero_zero_int=1. member_int=1. one_one_int=1. minus_minus_int=1. times_times_int=1. plus_plus_int=1. zero_zero_real=1. zero_zero_nat=1. bit1=1. pls=1. ord_less_real=1. ord_less_eq_real=1. bit0=1. ord_less_eq_int=1. ord_less_nat=1. times_times_real=1. number_number_of_nat=1. times_times_nat=1. suc=1. number_number_of_int=1. plus_plus_nat=1. ord_less_eq_nat=1. plus_plus_real=1. power_power_real=1. number267125858f_real=1. dvd_dvd_nat=1. one_one_real=1. zero_zero_rat=1. one_one_nat=1. times_times_rat=1. dvd_dvd_int=1. inverse_divide_real=1. power_power_int=1. div_div_int=1. nat=1. real_nat=1. semiri1621563631at_int=1. minus_minus_nat=1. ord_less_rat=1. ord_less_eq_rat=1. minus_minus_real=1. plus_plus_rat=1. times_times_complex=1. power_power_nat=1. zero_z126310315umeral=1. times_123202395de_int=1. times_1655362735umeral=1. zero_z891286103de_int=1. min=1. cos=1. div_div_nat=1. gcd_gcd_nat=1. plus_p1446045655de_int=1. plus_p1627245867umeral=1. plus_plus_complex=1. gcd_gcd_int=1. cOMBS_nat_real_real=1. power_power_rat=1. abs_abs_real=1. cOMBB_1308947378al_nat=1. pi=1. product_Pair_int_int=1. sin=1. fact_fact_nat=1. real_int=1. ord_le1860547276de_int=1. uminus_uminus_real=1. ord_le1304079648umeral=1. uminus_uminus_int=1. ord_le258702272de_int=1. ord_le565307924umeral=1. even_odd_even_nat=1. power_2100829034umeral=1. power_881366806de_int=1. number_number_of_rat=1. abs_abs_int=1. sqrt=1. root=1. one_one_rat=1. produc865579683l_real=1. cOMBB_nat_real_nat=1. zero_zero_complex=1. number528085621omplex=1. power_power_complex=1. fract=1. big_co604158596t_real=1. nat_neg=1. even_odd_even_int=1. ln=1. complex=1. re=1. dvd_dv174992974umeral=1. dvd_dv1760642554de_int=1. im=1. zprime=1. coprime=1. fconj=1. minus_minus_rat=1. one_on1645066479umeral=1. inverse_inverse_real=1. minus_minus_complex=1. one_on1684967323de_int=1. sequentially=1. powr=1. cnj=1. dvd_dvd_complex=1. dvd_dvd_rat=1. dvd_dvd_real=1. exp_real=1. cOMBC_nat_nat_nat=1. div_di1430059507de_int=1. div_di1218280263umeral=1. tan=1. prime=1. zEven=1. cOMBC_nat_real_real=1. one_one_complex=1. pred=1. zOdd=1. semiri132038758t_real=1. semiri151668891at_rat=1. product_snd_int_int=1. suminf_real=1. cOMBB_1652995168ol_int=1. cOMBB_882763271l_real=1. cOMBS_int_bool_bool=1. cOMBB_real_real_nat=1. semiri984289939at_nat=1. norm_norm_complex=1. semiri1424489471de_int=1. semiri1619134803umeral=1. nat_nat_set=1. finite_finite_nat=1. nat_is_nat=1. member_real=1. product_Pair_nat_nat=1. member_nat=1. arctan=1. product_fst_int_int=1. number1226105091de_int=1. product_snd_nat_nat=1. semiri2020571505omplex=1. finite_card_int=1. m=1. ii=1. number1443263063umeral=1. cOMBS_real_real_real=1. fact_fact_int=1. cOMBC_int_int_bool=1. cOMBB_963856155at_nat=1. finite_finite_int=1. cOMBC_real_nat_real=1. arccos=1. produc1935615926l_real=1. produc556554744l_real=1. fequal_int=1. arcsin=1. cOMBC_nat_nat_bool=1. sRStar=1. finite_card_nat=1. cOMBB_624415285l_real=1. cOMBC_real_real_real=1. cOMBS_real_bool_bool=1. quadRes=1. cOMBB_599996122l_real=1. cOMBC_real_real_bool=1. fFalse=1. fTrue=1. member_fun_int_bool=1. quickcheck_of_int=1. fequal_real=1. ord_min_nat=1. product_fst_nat_nat=1. cOMBB_real_real_real=1. cOMBC_int_int_int=1. norRRset=1. ord_lessThan_nat=1. ord_max_nat=1. s=1. succ=1. cOMBI_int=1. if_real=1. t=1. uminus473333897omplex=1. cOMBB_64450052al_nat=1. cOMBB_722610049al_nat=1. invers1449016382omplex=1. negateSnd=1. ord_le1568362934t_bool=1. sgn_sgn_int=1. sin_coeff=1. cOMBB_1480366270al_nat=1. cOMBB_782625832nt_int=1. cOMBB_nat_nat_nat=1. cos_coeff=1. if_nat=1. norm_frac_rel=1. sgn_sgn_real=1. abs_abs_rat=1. cOMBB_1419615765al_nat=1. cOMBB_1948233853l_real=1. cOMBB_430461877l_real=1. fact=1. if_Pro1731782967nt_int=1. inverse_divide_rat=1. negDivAlg_rel=1. posDivAlg_rel=1. ring_1_of_int_rat=1. big_co1758102091ol_nat=1. cOMBB_1496585939nt_int=1. cOMBB_1934313333l_real=1. cOMBC_2010481033l_real=1. if_int=1. twoSqu1770640450sum2sq=1. cOMBB_1015721476ol_nat=1. cOMBB_800536526ol_nat=1. cOMBS_nat_bool_bool=1. field_1210416355s_real=1. norm_norm_real=1. ord_atMost_nat=1. produc282740534nt_int=1. s1=1. twoSqu1907779896sum2sq=1. uminus_uminus_rat=1. zfact=1. big_co387207925at_nat=1. cOMBB_117013006ex_nat=1. cOMBS_337378985l_real=1. d22set=1. fEx_int=1. fNot=1. int_gcd=1. invers1025623611omplex=1. lazy_small_lazy_rel=1. member180897546at_nat=1. nat_gcd_rel=1. ord_le951220754t_bool=1. ring_1_of_int_real=1. sgn_sgn_complex=1. upto_rel=1. cOMBB_1533963633al_nat=1. cOMBB_1550063917nt_int=1. cOMBB_182850971nt_int=1. cOMBB_662120866nt_int=1. cOMBB_bool_bool_int=1. cOMBB_int_bool_int=1. cOMBC_226598744l_bool=1. cOMBC_746811033l_real=1. cOMBS_1470252563nt_int=1. finite1395289673t_bool=1. ord_le1912455174t_bool=1. ord_le382113706t_bool=1. phi=1. quickc1265749348ro_rel=1. ring_11397209091omplex=1. sgn_sgn_rat=1. cOMBB_1344650379nt_int=1. cOMBB_1418110531ol_int=1. cOMBB_1506897658nt_int=1. cOMBB_1768400739l_real=1. cOMBB_1846624359nt_int=1. cOMBB_1896127834l_real=1. cOMBB_1926947at_nat=1. cOMBB_2092576745l_real=1. cOMBB_320287386nt_int=1. cOMBB_421464275l_real=1. cOMBB_638531587nt_int=1. cOMBC_1329014627t_real=1. cOMBC_1508019322l_real=1. cOMBC_1805044090nt_int=1. cOMBC_1964283556nt_int=1. cOMBC_33652895t_bool=1. cOMBC_707974837nt_int=1. cOMBC_94739984l_bool=1. cOMBI_nat=1. cOMBS_1891347093nt_int=1. cOMBS_400904860l_bool=1. cOMBS_nat_rat_rat=1. fequal_nat=1. inverse_inverse_rat=1. member2143287562nt_int=1. produc494345619at_nat=1. size_size_nat=1. suminf_complex=1. big_co230513141nt_int=1. cOMBB_1121934354l_real=1. cOMBB_1146692694ol_nat=1. cOMBB_1484610687nt_int=1. cOMBB_1946221070al_int=1. cOMBB_2027861294ol_int=1. cOMBB_2107848501l_real=1. cOMBB_389152643ol_int=1. cOMBB_422423973l_real=1. cOMBB_591320580ol_int=1. cOMBB_793166024ol_int=1. cOMBB_real_bool_int=1. cOMBB_real_bool_real=1. cOMBC_1683390479t_bool=1. cOMBC_983868720t_bool=1. cOMBC_int_nat_int=1. cOMBS_193881978l_real=1. fequal_rat=1. if_Pro313124157l_real=1. member308914164t_bool=1. minus_1690775515umeral=1. nat_size=1. ratrel=1. size_s945831648umeral=1. sr=1. cOMBB_1102189010nt_int=1. cOMBB_1159313332l_real=1. cOMBB_118231410ol_int=1. cOMBB_1211207212ol_int=1. cOMBB_1247074793l_real=1. cOMBB_1290694363ol_int=1. cOMBB_1317819104ol_int=1. cOMBB_1389251822nt_int=1. cOMBB_1453147152l_real=1. cOMBB_1459832681l_real=1. cOMBB_1593174431ol_int=1. cOMBB_1647910634ol_int=1. cOMBB_1657320178nt_int=1. cOMBB_170828966nt_int=1. cOMBB_1714920583ex_nat=1. cOMBB_172850899l_real=1. cOMBB_1795223523l_real=1. cOMBB_1875755114nt_int=1. cOMBB_188663909l_real=1. cOMBB_2046938128ol_int=1. cOMBB_2050079537l_real=1. cOMBB_2086486755nt_int=1. cOMBB_2087181719ol_int=1. cOMBB_2125215742l_real=1. cOMBB_2129825580al_int=1. cOMBB_2138578194l_real=1. cOMBB_214503962ol_int=1. cOMBB_222539774nt_int=1. cOMBB_237479429l_real=1. cOMBB_270132509l_real=1. cOMBB_348438404at_nat=1. cOMBB_3698607l_real=1. cOMBB_415306361l_real=1. cOMBB_421479067l_real=1. cOMBB_44084907l_real=1. cOMBB_47643171nt_int=1. cOMBB_520243667l_real=1. cOMBB_550562460ol_int=1. cOMBB_557258983l_real=1. cOMBB_562416518ol_int=1. cOMBB_563965291nt_int=1. cOMBB_569260360nt_int=1. cOMBB_595786433nt_int=1. cOMBB_650041899nt_int=1. cOMBB_690122963l_real=1. cOMBB_731595089l_real=1. cOMBB_735549356nt_int=1. cOMBB_773384995nt_int=1. cOMBB_798864284nt_int=1. cOMBB_859312247nt_nat=1. cOMBB_866192175nt_int=1. cOMBB_929094628al_int=1. cOMBB_95956018ol_int=1. cOMBB_984071017l_real=1. cOMBB_bool_bool_nat=1. cOMBB_int_bool_nat=1. cOMBB_int_int_int=1. cOMBB_int_int_nat=1. cOMBB_int_rat_int=1. cOMBB_int_real_int=1. cOMBB_nat_bool_int=1. cOMBB_rat_bool_int=1. cOMBB_rat_rat_nat=1. cOMBB_real_bool_nat=1. cOMBB_real_real_int=1. cOMBC_1113900202t_bool=1. cOMBC_1417754532nt_int=1. cOMBC_1801450329t_bool=1. cOMBC_2037038898nt_int=1. cOMBC_241247961t_bool=1. cOMBC_243356514t_bool=1. cOMBC_295337211t_real=1. cOMBC_412863505t_bool=1. cOMBC_467724138l_real=1. cOMBC_678732631t_bool=1. cOMBC_736024425nt_int=1. cOMBC_824100275t_real=1. cOMBC_int_rat_bool=1. cOMBC_int_real_bool=1. cOMBC_nat_int_int=1. cOMBI_real=1. cOMBS_107495110l_real=1. cOMBS_1610675060t_bool=1. cOMBS_163651580l_real=1. cOMBS_1640193733nt_int=1. cOMBS_1676841879t_bool=1. cOMBS_1720984575t_bool=1. cOMBS_2011279539nt_int=1. cOMBS_2019229620nt_int=1. cOMBS_480885903t_bool=1. cOMBS_511527878l_real=1. cOMBS_549334880nt_rat=1. cOMBS_696648398omplex=1. cOMBS_794971624l_real=1. cOMBS_902912708l_real=1. cOMBS_nat_nat_nat=1. fEx_nat=1. iszero_int=1. iszero_rat=1. pair_leq=1. pair_less=1. produc883642259nt_int=1. ring_1_Ints_real=1. ring_1_of_int_int=1. big_co1024481617at_int=1. finite1876863882t_bool=1. frac=1. int=1. minus_534354567de_int=1. ord_lessThan_real=1. pred_nat=1. produc1318306967de_int=1. produc2136830103umeral=1. produc713279741t_bool=1. size_size_complex=1. size_size_list_int=1. c1=1. c2=1. c3=1. c4=1. c5=1. c6=1. c7=1. c8=1. c9=1. c10=1. c11=1. c12=1. c13=1. c14=1. c15=1. c16=1. c17=1. c18=1. hAPP_int_bool=1. hAPP_int_int=1. hAPP_i1948725293t_bool=1. hAPP_int_fun_int_int=1. hAPP_f448129468l_bool=1. hAPP_i2112223885l_bool=1. hAPP_real_real=1. hAPP_nat_nat=1. hAPP_real_bool=1. hAPP_r1134773055l_bool=1. hAPP_nat_bool=1. hAPP_nat_fun_nat_nat=1. hAPP_n1699378549t_bool=1. hAPP_r1250527377l_real=1. hAPP_int_nat=1. hAPP_nat_real=1. hAPP_rat_rat=1. hAPP_rat_fun_rat_rat=1. hAPP_nat_int=1. hAPP_rat_bool=1. hAPP_r2115560837t_bool=1. hAPP_int_real=1. hAPP_complex_complex=1. hAPP_r474017924t_real=1. hAPP_Q1762011733de_int=1. hAPP_C498520661umeral=1. hAPP_c172932010omplex=1. hAPP_Q1729056540de_int=1. hAPP_C1594335432umeral=1. hAPP_Q2096512830t_bool=1. hAPP_C24501650l_bool=1. hAPP_Q1010094925t_bool=1. hAPP_C10902773l_bool=1. hAPP_int_fun_nat_int=1. hAPP_n546711566l_real=1. hAPP_int_rat=1. hAPP_nat_rat=1. hAPP_f1888375463t_real=1. hAPP_complex_real=1. hAPP_f451358433l_real=1. hAPP_n1108039445umeral=1. hAPP_n522471361de_int=1. hAPP_f197615720t_real=1. hAPP_rat_fun_nat_rat=1. hAPP_f1092742621l_real=1. hAPP_i1584592887nt_int=1. hAPP_i1524277240nt_int=1. hAPP_C471253896umeral=1. hAPP_Q125967240de_int=1. deriv_real=1. hAPP_nat_complex=1. hAPP_r2019696015l_real=1. hAPP_r1195171167l_real=1. hAPP_P1333854989l_bool=1. hAPP_int_complex=1. hAPP_f1566856189t_real=1. hAPP_f1676298106t_real=1. hAPP_c1088319240omplex=1. hAPP_int_fun_int_rat=1. hAPP_f54304608l_bool=1. hAPP_P1175774780nt_int=1. hAPP_f1509287679l_real=1. hAPP_f1406902706l_real=1. hAPP_P603027463t_bool=1. hAPP_r265291036omplex=1. hAPP_real_complex=1. hAPP_complex_bool=1. hAPP_c1403687985x_bool=1. hAPP_f416620757at_nat=1. hAPP_f42270917t_real=1. hAPP_f1805168059t_bool=1. hAPP_P1654798560at_nat=1. hAPP_i1732201573de_int=1. hAPP_f1950183573l_real=1. hAPP_f352196356l_real=1. hAPP_f1734373249l_bool=1. hAPP_f1960414057l_real=1. hAPP_f2144054103l_bool=1. hAPP_f1690286043t_real=1. hAPP_f727283836t_bool=1. hAPP_P1555980039t_bool=1. hAPP_P1975530577nt_int=1. hAPP_f556082547l_real=1. hAPP_n332733730l_real=1. hAPP_f1501066425l_bool=1. hAPP_n1865633855at_nat=1. hAPP_r1130899993l_bool=1. hAPP_P731461727l_real=1. hAPP_n1289843868at_nat=1. hAPP_f428220345t_bool=1. hAPP_n215258509l_bool=1. divmod_nat_rel=1. posDivAlg=1. divmod_int_rel=1. negDivAlg=1. hAPP_i769753017umeral=1. hAPP_f2069516469l_real=1. hAPP_f957591787ol_nat=1. hAPP_f1639111240at_nat=1. hAPP_f1731313045at_nat=1. hAPP_f22106695ol_nat=1. image_nat_int=1. hAPP_f248847589l_real=1. image_int_nat=1. hAPP_f215623910l_bool=1. hAPP_bool_bool=1. hAPP_f229349961t_bool=1. rcis=1. hAPP_f1010710745l_bool=1. hAPP_f103356543l_bool=1. hAPP_f1485493409l_bool=1. hAPP_f1574055885nt_int=1. hAPP_f1861673493l_bool=1. hAPP_f60716669l_bool=1. big_co1548731110nt_int=1. divmod_int=1. hAPP_f1407095736t_real=1. hAPP_f1796904975t_real=1. hAPP_f628503027l_bool=1. hAPP_f695504873l_bool=1. hAPP_f203520653l_real=1. hAPP_f800510211t_bool=1. divmod_nat=1. hAPP_f1992866895l_real=1. hAPP_f2135977429nt_int=1. bezw=1. hAPP_f1914919701at_nat=1. image_int_int=1. hAPP_f1290670424t_real=1. hAPP_f284875647l_bool=1. hAPP_f327026105t_real=1. hAPP_f73570213t_real=1. hAPP_f918606171t_real=1. hAPP_b992065680at_nat=1. hAPP_f1129044203l_real=1. hAPP_f1585078997at_nat=1. hAPP_f1760145644nt_int=1. hAPP_f2054677283nt_int=1. hAPP_f264196187l_real=1. norm_frac=1. setS=1. hAPP_b589554111l_bool=1. hAPP_f1393328161l_bool=1. hAPP_f1900646853l_bool=1. hAPP_f1904307534l_real=1. hAPP_f1993765077t_real=1. hAPP_f224989205l_real=1. hAPP_f465821005nt_int=1. hAPP_f521865025ol_nat=1. hAPP_f895854793t_real=1. rRset2norRR=1. hAPP_b1463609396nt_int=1. hAPP_f1067681227l_real=1. hAPP_f1523861863l_real=1. hAPP_f1629352853nt_int=1. hAPP_f1644353557l_real=1. hAPP_f2119584768l_real=1. hAPP_f237669397t_bool=1. hAPP_f59594445l_real=1. hAPP_f93439879ol_nat=1. hAPP_r337325687l_real=1. hAPP_P1145851913nt_int=1. hAPP_P408881810nt_int=1. hAPP_f1080886329l_bool=1. hAPP_f1146629647l_bool=1. hAPP_f115998247l_bool=1. hAPP_f1505651103t_bool=1. hAPP_f561022312t_bool=1. hAPP_f618557131t_bool=1. hAPP_i1730167831nt_int=1. pdivmod=1. hAPP_P1876494581l_bool=1. hAPP_b40753821nt_int=1. hAPP_f1457396693omplex=1. hAPP_f1646117269omplex=1. hAPP_f2062659861at_rat=1. hAPP_f335634176l_real=1. hAPP_f782000547ol_nat=1. hAPP_i345030060t_bool=1. fdisj=1. hAPP_P989584703t_bool=1. hAPP_f1081447804nt_int=1. hAPP_f1607377819t_real=1. hAPP_f1722879237t_bool=1. hAPP_f1791153283t_bool=1. hAPP_f1961001761l_bool=1. hAPP_f2126667875l_real=1. hAPP_f214467655t_bool=1. hAPP_f251264923nt_int=1. hAPP_f279307923nt_int=1. hAPP_f291087797t_bool=1. hAPP_f292751352t_real=1. hAPP_f365317245l_bool=1. hAPP_f526364711l_bool=1. hAPP_f627970963t_bool=1. hAPP_nat_fun_int_int=1. hAPP_r195310020l_real=1. hAPP_r481142414t_bool=1. nat_case_nat=1. hAPP_P1062890741l_bool=1. hAPP_P1668131738nt_int=1. hAPP_P1860904029l_real=1. hAPP_P2110489235nt_int=1. hAPP_P369611207at_nat=1. hAPP_P402336767at_nat=1. hAPP_b646336293l_real=1. hAPP_f1038672605nt_int=1. hAPP_f1073644437at_rat=1. hAPP_f1122023986l_real=1. hAPP_f1131305186at_rat=1. hAPP_f1135648319t_real=1. hAPP_f1163894827t_real=1. hAPP_f1310479593l_real=1. hAPP_f1377453843nt_int=1. hAPP_f142232355nt_int=1. hAPP_f1469637905l_bool=1. hAPP_f1501741374t_real=1. hAPP_f1512942609t_bool=1. hAPP_f1582586579nt_int=1. hAPP_f1591894897l_real=1. hAPP_f1594865479ol_int=1. hAPP_f167782601nt_int=1. hAPP_f1732848529l_real=1. hAPP_f1781059817t_bool=1. hAPP_f1786887945l_real=1. hAPP_f1830834240at_rat=1. hAPP_f1855025978nt_int=1. hAPP_f1874145443l_real=1. hAPP_f197974549nt_int=1. hAPP_f1998755145t_real=1. hAPP_f2018027565t_bool=1. hAPP_f202917053t_bool=1. hAPP_f2119767738t_bool=1. hAPP_f213450978omplex=1. hAPP_f423804115t_bool=1. hAPP_f429935748t_bool=1. hAPP_f461489489t_bool=1. hAPP_f513299955t_bool=1. hAPP_f521271395nt_int=1. hAPP_f617310012nt_int=1. hAPP_f617735571l_real=1. hAPP_f791698359t_bool=1. hAPP_f892584630t_bool=1. hAPP_f960630521nt_int=1. hAPP_i119638232t_bool=1. hAPP_i1216375074nt_int=1. hAPP_i1529485324t_bool=1. hAPP_i1778081398nt_int=1. hAPP_i418383825t_bool=1. hAPP_r373117745t_bool=1. hAPP_r632163725t_bool=1. hAPP_r712953929l_real=1. image_nat_nat=1. xzgcd=1. hAPP_C2129356693al_nat=1. hAPP_P1463840893t_bool=1. hAPP_P1681069257l_bool=1. hAPP_P1982799375l_real=1. hAPP_P252494982t_bool=1. hAPP_P621635040nt_int=1. hAPP_P62333565t_bool=1. hAPP_P880235623l_bool=1. hAPP_P921862901l_bool=1. hAPP_b1482221219l_real=1. hAPP_f101635158l_bool=1. hAPP_f1048215610t_bool=1. hAPP_f1119546819l_real=1. hAPP_f1148171384nt_int=1. hAPP_f1173902867t_bool=1. hAPP_f1251097615t_bool=1. hAPP_f1344929775l_bool=1. hAPP_f1352333353l_real=1. hAPP_f1402056515l_bool=1. hAPP_f1475822739t_bool=1. hAPP_f1545556668t_bool=1. hAPP_f1735705551nt_int=1. hAPP_f1830214065l_bool=1. hAPP_f1925507189l_bool=1. hAPP_f1926459811ol_int=1. hAPP_f2037783381l_bool=1. hAPP_f2105620693nt_int=1. hAPP_f318350629l_real=1. hAPP_f320596107l_real=1. hAPP_f334321603nt_int=1. hAPP_f429384717t_bool=1. hAPP_f472159229t_bool=1. hAPP_f555557763l_real=1. hAPP_f654702867t_bool=1. hAPP_f66927821l_bool=1. hAPP_f735247205t_bool=1. hAPP_f824790735l_bool=1. hAPP_f850851693l_real=1. hAPP_f85237525t_bool=1. hAPP_i1153444985nt_int=1. hAPP_i1836591008nt_int=1. hAPP_i2104874254nt_int=1. hAPP_i371647666l_real=1. hAPP_i68813070l_bool=1. hAPP_n682674106l_real=1. big_co1705425894at_nat=1. hAPP_P1400155668t_bool=1. hAPP_P1853363688nt_rat=1. hAPP_P2053772734t_bool=1. hAPP_P230595783nt_int=1. hAPP_P507092031nt_rat=1. hAPP_P752050971nt_int=1. hAPP_P877703507t_bool=1. hAPP_P941956607nt_int=1. hAPP_f1040184293l_real=1. hAPP_f1052840010omplex=1. hAPP_f1063001928t_bool=1. hAPP_f1065357839nt_rat=1. hAPP_f1076489843l_real=1. hAPP_f1082758869at_rat=1. hAPP_f1083049571t_bool=1. hAPP_f1084724746t_bool=1. hAPP_f1108539711l_real=1. hAPP_f1109256153l_real=1. hAPP_f11151005nt_int=1. hAPP_f1115375056t_bool=1. hAPP_f1120137083nt_rat=1. hAPP_f1121609875t_bool=1. hAPP_f1131424617t_bool=1. hAPP_f1139079189at_int=1. hAPP_f1151807883nt_int=1. hAPP_f1154658180t_bool=1. hAPP_f1161935955l_real=1. hAPP_f1169087445t_real=1. hAPP_f11709033t_bool=1. hAPP_f1177545951t_bool=1. hAPP_f1182713365t_bool=1. hAPP_f1187443546t_bool=1. hAPP_f1198736473t_bool=1. hAPP_f1222178131t_bool=1. hAPP_f1228585167nt_int=1. hAPP_f1241350704t_bool=1. hAPP_f125788821nt_int=1. hAPP_f125837377t_bool=1. hAPP_f1261108921t_bool=1. hAPP_f1270196437nt_rat=1. hAPP_f1273270095t_bool=1. hAPP_f1293550189l_real=1. hAPP_f1295083291t_real=1. hAPP_f1311564564nt_int=1. hAPP_f1315737299t_bool=1. hAPP_f1323932638t_real=1. hAPP_f1325804733t_bool=1. hAPP_f1341793266t_bool=1. hAPP_f134192053t_real=1. hAPP_f1356864277nt_int=1. hAPP_f1361476881t_bool=1. hAPP_f1375227553nt_int=1. hAPP_f1399575567t_bool=1. hAPP_f1408247010at_nat=1. hAPP_f1431025877at_int=1. hAPP_f1448486952t_bool=1. hAPP_f1456711209t_bool=1. hAPP_f1459420095t_bool=1. hAPP_f1480506641t_bool=1. hAPP_f151095827nt_int=1. hAPP_f1547007718nt_rat=1. hAPP_f1576199059t_bool=1. hAPP_f1607917947l_real=1. hAPP_f1613846310t_real=1. hAPP_f1625314454t_bool=1. hAPP_f1663701695l_real=1. hAPP_f1673103573at_int=1. hAPP_f1720068861t_real=1. hAPP_f172451459l_bool=1. hAPP_f1730463541l_real=1. hAPP_f1731308757nt_int=1. hAPP_f1736103445l_real=1. hAPP_f1763467369l_real=1. hAPP_f1769710159nt_int=1. hAPP_f1815301987t_bool=1. hAPP_f1832844496t_bool=1. hAPP_f183342685l_real=1. hAPP_f1840085295t_bool=1. hAPP_f1845081853t_bool=1. hAPP_f1878066172t_bool=1. hAPP_f1879982819l_real=1. hAPP_f1884005689l_bool=1. hAPP_f1900950721nt_rat=1. hAPP_f1902329768t_bool=1. hAPP_f1908301369l_real=1. hAPP_f1914742907nt_int=1. hAPP_f1920544743t_bool=1. hAPP_f1927674023t_bool=1. hAPP_f1935805932nt_int=1. hAPP_f1965248563t_bool=1. hAPP_f1990636069t_bool=1. hAPP_f2002277285nt_int=1. hAPP_f2004554147nt_int=1. hAPP_f2042071795t_bool=1. hAPP_f2050994651t_bool=1. hAPP_f2072293513l_real=1. hAPP_f2084827628t_bool=1. hAPP_f2094497995nt_int=1. hAPP_f2114902373l_real=1. hAPP_f212164053t_real=1. hAPP_f219720582omplex=1. hAPP_f223623489t_bool=1. hAPP_f23851874omplex=1. hAPP_f252743847l_bool=1. hAPP_f269654879t_real=1. hAPP_f271976045t_bool=1. hAPP_f289318786t_bool=1. hAPP_f301246617nt_int=1. hAPP_f324002831l_real=1. hAPP_f326728547t_bool=1. hAPP_f328572565l_real=1. hAPP_f330985925l_bool=1. hAPP_f332806115t_bool=1. hAPP_f340152631t_bool=1. hAPP_f351634443t_bool=1. hAPP_f352226557t_real=1. hAPP_f374502857t_bool=1. hAPP_f381769257l_bool=1. hAPP_f382317021nt_rat=1. hAPP_f389116883t_bool=1. hAPP_f400200275nt_int=1. hAPP_f400538607nt_int=1. hAPP_f403082607t_bool=1. hAPP_f406530075omplex=1. hAPP_f41603166t_bool=1. hAPP_f422375781t_bool=1. hAPP_f434654626nt_int=1. hAPP_f492612297t_bool=1. hAPP_f499530369l_bool=1. hAPP_f512004221l_bool=1. hAPP_f517738447l_real=1. hAPP_f534361363nt_int=1. hAPP_f538565589t_bool=1. hAPP_f55007289nt_int=1. hAPP_f563293398t_real=1. hAPP_f571820015t_bool=1. hAPP_f578362681nt_int=1. hAPP_f601898971t_bool=1. hAPP_f605995867t_bool=1. hAPP_f643658017l_bool=1. hAPP_f644105375l_real=1. hAPP_f659380387ol_int=1. hAPP_f677014621l_real=1. hAPP_f693283245t_bool=1. hAPP_f696844121t_bool=1. hAPP_f704188951t_bool=1. hAPP_f713949180nt_int=1. hAPP_f720654810t_bool=1. hAPP_f725770969t_bool=1. hAPP_f729588221t_bool=1. hAPP_f737665877t_bool=1. hAPP_f74147283t_real=1. hAPP_f743790483t_bool=1. hAPP_f747162069nt_int=1. hAPP_f747475781nt_int=1. hAPP_f766218899t_real=1. hAPP_f77155443t_bool=1. hAPP_f774948957l_real=1. hAPP_f800361423l_real=1. hAPP_f801032407l_bool=1. hAPP_f808647893nt_int=1. hAPP_f823702159l_real=1. hAPP_f834119977l_bool=1. hAPP_f836602773nt_int=1. hAPP_f837737749at_nat=1. hAPP_f847817363t_bool=1. hAPP_f848516149l_bool=1. hAPP_f857296063t_bool=1. hAPP_f866672739t_bool=1. hAPP_f883232462nt_int=1. hAPP_f888249301nt_int=1. hAPP_f894608603t_bool=1. hAPP_f908327869t_bool=1. hAPP_f940061907l_bool=1. hAPP_f956464319t_bool=1. hAPP_f983569339t_bool=1. hAPP_f999082675at_nat=1. hAPP_i157317923t_real=1. hAPP_i16292313t_bool=1. hAPP_i1671359920t_real=1. hAPP_i1796122964t_bool=1. hAPP_i1823918693l_bool=1. hAPP_i447943416t_real=1. hAPP_i61245889nt_int=1. hAPP_i614188481l_bool=1. hAPP_i627165055t_real=1. hAPP_i688218208l_bool=1. hAPP_i744251628nt_int=1. hAPP_i872162435t_bool=1. hAPP_i998473677nt_int=1. hAPP_n1006566506l_bool=1. hAPP_n1983812447omplex=1. hAPP_nat_fun_rat_rat=1. hAPP_r1043989325l_real=1. hAPP_r1150855451l_bool=1. hAPP_r1230691443l_real=1. hAPP_r1392521531t_bool=1. hAPP_r1409264866l_real=1. hAPP_r1562208522t_bool=1. hAPP_r1623423338t_bool=1. hAPP_r1900510169t_bool=1. hAPP_r1938957037l_bool=1. hAPP_r1962736578t_bool=1. hAPP_r2052264067t_bool=1. hAPP_r224952523l_real=1. hAPP_r279757445t_bool=1. hAPP_r285062775l_bool=1. hAPP_r50993370t_real=1. hAPP_r579619957l_real=1. hAPP_r653725362t_bool=1. hAPP_r794344869l_real=1. hAPP_r847664445t_bool=1. hAPP_r931665397l_real=1. hAPP_r938550413l_real=1. multInvPair=1. nat_case_bool=1. code_d418564891umeral=1. hAPP_C1614127039umeral=1. hAPP_C2132160860umeral=1. hAPP_P1586233937at_nat=1. hAPP_Q1256397320de_int=1. hAPP_Q952608535de_int=1. hAPP_complex_nat=1. hAPP_f1418982953t_bool=1. hAPP_f1599440987ol_int=1. hAPP_f538843494t_bool=1. hAPP_list_int_nat=1. image_275383677t_bool=1. quickc495462417de_int=1. f2=1. f16=1. f17=1. f28=1. f29=1. f31=1. f39=1. f41=1. f42=1. f53=1. f56=1. f62=1. f63=1. f69=1. f73=1. f75=1. f76=1. f77=1. f78=1. f79=1. f81=1. f82=1. f86=1. f92=1. f93=1. f95=1. f96=1. f98=1. f99=1. f100=1. f101=1. f117=1. f125=1. f136=1. f137=1. f138=1. f146=1. f150=1. f151=1. f164=1. f168=1. f171=1. f175=1. f176=1. f187=1. f191=1. f192=1. f194=1. f195=1. f205=1. f212=1. f215=1. f219=1. f220=1. f232=1. f233=1. f238=1. f242=1. f243=1. f245=1. f247=1. f261=1. f263=1. f264=1. f266=1. f269=1. f270=1. f272=1. f274=1. f278=1. f279=1. f287=1. f300=1. f302=1. f303=1. f304=1. f305=1. f306=1. f307=1. f308=1. f309=1. f310=1. f311=1. f320=1. f327=1. f329=1. f335=1. f336=1. f337=1. f338=1. f339=1. f340=1. f341=1. f346=1. f347=1. f353=1. f359=1. f361=1. f367=1. f368=1. f373=1. f375=1. f377=1. f378=1. f379=1. f381=1. f382=1. f388=1. f401=1. f402=1. f408=1. f409=1. f411=1. f412=1. f413=1. f414=1. f427=1. f428=1. f429=1. f432=1. f433=1. f434=1. f435=1. f436=1. f440=1. f441=1. f443=1. f447=1. f448=1. f450=1. f451=1. f455=1. f457=1. f458=1. f461=1. f462=1. f465=1. f466=1. f467=1. f470=1. f476=1. f486=1. f488=1. f490=1. f492=1. f497=1. f502=1. f513=1. f514=1. ord_at875362053st_int=1. div_mod_int=1. zcong=1. div_mod_nat=1. legacy_zgcd=1. ord_at4362885an_nat=1. div_mo231679042de_int=1. div_mo1740067990umeral=1. isCont_real_real=1. accp_P2006205492nt_int=1. archim1246769320r_real=1. natfloor=1. quickcheck_int_of=1. multInv=1. ratreal=1. summable_real=1. log=1. of_real_complex=1. collect_int=1. archim856651990g_real=1. natceiling=1. quotient_of=1. cis=1. standardRes=1. resSet=1. vanishes=1. ord_at238088361st_nat=1. collect_nat=1. is_RRset=1. wset=1. normalize=1. sums_real=1. adjust=1. comple124823625p_real=1. monoseq_real=1. bnorRset=1. ord_gr788844697n_real=1. inv=1. nat_gcd=1. ord_at641636577an_int=1. scaleR1652505878omplex=1. code_S1047413653umeral=1. summable_complex=1. the_real=1. at_real=1. cauchy_real=1. expi=1. frct=1. legendre=1. nat_tsub=1. accp_P490777396at_nat=1. code_int_of=1. produc1518849193nt_int=1. rsetR=1. accp_int=1. archim791455193or_rat=1. cOMBK_1643584600t_bool=1. cauchy_complex=1. diffs_real=1. fimplies=1. arg=1. comple219730294t_bool=1. hilbert_Eps_int=1. produc93441119t_bool=1. sums_complex=1. at_complex=1. bseq_real=1. cOMBK_rat_nat=1. code_c271388182l_size=1. code_nat_of_aux=1. collec1347809874nt_int=1. collect_real=1. nat_tr160667106at_int=1. nat_tr876908586nt_nat=1. noXRRset=1. ord_at1589558736t_real=1. ord_gr1297742076an_int=1. ord_gr660468384an_nat=1. produc713050258nt_int=1. scaleR_scaleR_real=1. the_int=1. zcongm=1. cOMBK_bool_nat=1. collec1979865426at_nat=1. collec50511176nt_int=1. isCont_complex_real=1. nat_lcm=1. produc1298267108nt_int=1. z3div=1. z3mod=1. bijR_int_int=1. complex_size=1. dist_dist_complex=1. dist_dist_real=1. hilbert_Eps_real=1. int_ge_less_than=1. int_ge_less_than2=1. int_lcm=1. isCont156215680omplex=1. nat_aux=1. ord_at1496968948n_real=1. produc1391996073at_nat=1. produc1922581855t_bool=1. produc595218619l_real=1. produc973645403t_bool=1. quickc666637781d_zero=1. quickcheck_nat_of=1. the_Pr2103884470nt_int=1. the_Pr588456374at_nat=1. trivial_limit_nat=1. undefined_int=1. f15=1. f18=1. f20=1. f23=1. f24=1. f25=1. f33=1. f34=1. f38=1. f43=1. f49=1. f51=1. f52=1. f61=1. f80=1. f87=1. f88=1. f89=1. f91=1. f94=1. f102=1. f105=1. f111=1. f112=1. f113=1. f114=1. f115=1. f116=1. f126=1. f144=1. f145=1. f149=1. f163=1. f186=1. f188=1. f190=1. f193=1. f196=1. f197=1. f198=1. f199=1. f200=1. f210=1. f211=1. f235=1. f236=1. f239=1. f240=1. f241=1. f244=1. f248=1. f252=1. f253=1. f258=1. f259=1. f260=1. f267=1. f268=1. f271=1. f291=1. f292=1. f316=1. f317=1. f328=1. f352=1. f355=1. f358=1. f360=1. f364=1. f370=1. f374=1. f376=1. f380=1. f383=1. f384=1. f387=1. f391=1. f416=1. f417=1. f426=1. f442=1. f446=1. f449=1. f456=1. f459=1. f460=1. f463=1. f464=1. f471=1. f472=1. f473=1. f474=1. f484=1. f487=1. f489=1. f491=1. f495=1. f496=1. f505=1. f506=1. f507=1. f508=1. f509=1. f519=1. f520=1. f523=1. f524=1. tendsto_nat_real=1. bolzano_bisect=1. tendsto_real_real=1. tendsto_nat_complex=1. tendsto_complex_real=1. tendst1507391555omplex=1. f13=1. f21=1. f22=1. f27=1. f30=1. f54=1. f64=1. f65=1. f66=1. f67=1. f68=1. f72=1. f74=1. f97=1. f103=1. f104=1. f106=1. f107=1. f109=1. f110=1. f127=1. f128=1. f132=1. f133=1. f135=1. f140=1. f141=1. f142=1. f143=1. f153=1. f157=1. f158=1. f159=1. f160=1. f165=1. f166=1. f172=1. f173=1. f174=1. f180=1. f181=1. f189=1. f206=1. f207=1. f208=1. f209=1. f213=1. f214=1. f217=1. f218=1. f221=1. f222=1. f231=1. f234=1. f237=1. f254=1. f255=1. f262=1. f273=1. f284=1. f285=1. f286=1. f288=1. f289=1. f290=1. f298=1. f299=1. f301=1. f312=1. f315=1. f318=1. f319=1. f324=1. f330=1. f334=1. f342=1. f345=1. f349=1. f363=1. f365=1. f366=1. f386=1. f392=1. f393=1. f394=1. f397=1. f398=1. f399=1. f400=1. f403=1. f404=1. f405=1. f415=1. f452=1. f454=1. f468=1. f469=1. f475=1. f483=1. f493=1. f494=1. f498=1. f499=1. f500=1. f501=1. f503=1. f504=1. f515=1. f516=1. f521=1. f522=1. f525=1. f526=1. f4=1. f5=1. f6=1. f7=1. f8=1. f9=1. f10=1. f11=1. f12=1. f19=1. f26=1. f36=1. f37=1. f40=1. f44=1. f45=1. f46=1. f47=1. f48=1. f58=1. f59=1. f60=1. f70=1. f71=1. f83=1. f84=1. f85=1. f90=1. f108=1. f119=1. f120=1. f121=1. f122=1. f123=1. f124=1. f129=1. f130=1. f131=1. f134=1. f147=1. f148=1. f152=1. f154=1. f155=1. f156=1. f161=1. f162=1. f169=1. f170=1. f177=1. f178=1. f179=1. f182=1. f183=1. f184=1. f216=1. f224=1. f225=1. f226=1. f227=1. f228=1. f229=1. f230=1. f249=1. f250=1. f251=1. f275=1. f276=1. f277=1. f280=1. f281=1. f296=1. f297=1. f313=1. f314=1. f322=1. f323=1. f331=1. f332=1. f333=1. f343=1. f344=1. f350=1. f351=1. f362=1. f371=1. f372=1. f385=1. f389=1. f390=1. f395=1. f396=1. f406=1. f407=1. f418=1. f419=1. f420=1. f421=1. f422=1. f423=1. f424=1. f425=1. f430=1. f431=1. f437=1. f438=1. f439=1. f444=1. f445=1. f477=1. f478=1. f479=1. f480=1. f3=1. f55=1. f57=1. f118=1. f201=1. f202=1. f203=1. f223=1. f256=1. f257=1. f293=1. f294=1. f356=1. f357=1. f481=1. f482=1. f510=1. f511=1. f512=1. f517=1. f518=1. f167=1. xzgcda=1. f325=1. f326=1. 12.40/12.39 12.40/12.39 ============================== STATISTICS ============================ 12.40/12.39 12.40/12.39 Given=0. Generated=8540. Kept=7709. proofs=0. 12.40/12.39 Usable=0. Sos=0. Demods=1074. Limbo=7708, Disabled=8540. Hints=0. 12.40/12.39 Kept_by_rule=0, Deleted_by_rule=0. 12.40/12.39 Forward_subsumed=831. Back_subsumed=0. 12.40/12.39 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 12.40/12.39 New_demodulators=1074 (16 lex), Back_demodulated=0. Back_unit_deleted=0. 12.40/12.39 Demod_attempts=376704. Demod_rewrites=2752. 12.40/12.39 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 12.40/12.39 Nonunit_fsub_feature_tests=3005. Nonunit_bsub_feature_tests=0. 12.40/12.39 Megabytes=419.43. 12.40/12.39 User_CPU=7.68, System_CPU=0.13, Wall_clock=8. 12.40/12.39 12.40/12.39 Megs malloced by palloc(): 400. 12.40/12.39 type (bytes each) gets frees in use bytes 12.40/12.39 chunk ( 104) 228464 228464 0 0.0 K 12.40/12.39 string_buf ( 8) 191942 191942 0 0.0 K 12.40/12.39 token ( 20) 500993 500993 0 0.0 K 12.40/12.39 pterm ( 16) 307803 307803 0 0.0 K 12.40/12.39 hashtab ( 8) 5134 5134 0 0.0 K 12.40/12.39 hashnode ( 8) 18521 18521 0 0.0 K 12.40/12.39 term ( 20) 2636708 2010687 626021 12227.0 K 12.40/12.39 term arg arrays: 3024.8 K 12.40/12.39 attribute ( 12) 46678 840 45838 537.2 K 12.40/12.39 ilist ( 8) 91870055 91860947 9108 71.2 K 12.40/12.39 plist ( 8) 147866 75601 72265 564.6 K 12.40/12.39 i2list ( 12) 965568 965568 0 0.0 K 12.40/12.39 just ( 12) 32796 6729 26067 305.5 K 12.40/12.39 parajust ( 16) 0 0 0 0.0 K 12.40/12.39 instancejust ( 8) 0 0 0 0.0 K 12.40/12.39 ivyjust ( 24) 0 0 0 0.0 K 12.40/12.39 formula ( 28) 196231 133807 62424 1706.9 K 12.40/12.39 formula arg arrays: 203.7 K 12.40/12.39 topform ( 52) 23844 831 23013 1168.6 K 12.40/12.39 clist_pos ( 20) 27492 8540 18952 370.2 K 12.40/12.39 clist ( 16) 8 0 8 0.1 K 12.40/12.39 context ( 808) 158471 158471 0 0.0 K 12.40/12.39 trail ( 12) 15156 15156 0 0.0 K 12.40/12.39 ac_match_pos (70044) 0 0 0 0.0 K 12.40/12.39 ac_match_free_vars_pos (20020) 12.40/12.39 0 0 0 0.0 K 12.40/12.39 btm_state ( 60) 0 0 0 0.0 K 12.40/12.39 btu_state ( 60) 0 0 0 0.0 K 12.40/12.39 ac_position (285432) 0 0 0 0.0 K 12.40/12.39 fpa_trie ( 20) 33426 0 33426 652.9 K 12.40/12.39 fpa_state ( 28) 1438 1438 0 0.0 K 12.40/12.39 fpa_index ( 12) 10 0 10 0.1 K 12.40/12.39 fpa_chunk ( 20) 29815 10444 19371 378.3 K 12.40/12.39 fpa_list ( 16) 12986 0 12986 202.9 K 12.40/12.39 fpa_list chunks: 789.5 K 12.40/12.39 discrim ( 12) 25094 0 25094 294.1 K 12.40/12.39 discrim_pos ( 16) 10818 10818 0 0.0 K 12.40/12.39 flat2 ( 32) 439707 439707 0 0.0 K 12.40/12.39 flat ( 48) 0 0 0 0.0 K 12.40/12.39 flatterm ( 32) 710339 710339 0 0.0 K 12.40/12.39 mindex ( 28) 13 0 13 0.4 K 12.40/12.39 mindex_pos ( 56) 128201 128201 0 0.0 K 12.40/12.39 lindex ( 12) 5 0 5 0.1 K 12.40/12.39 clash ( 40) 0 0 0 0.0 K 12.40/12.39 di_tree ( 12) 32921465 0 32921465 385798.4 K 12.40/12.39 avl_node ( 20) 0 0 0 0.0 K 12.40/12.39 12.40/12.39 Memory report, 20 @ 20 = 400 megs (400.00 megs used). 12.40/12.39 List 1, length 14, 0.1 K 12.40/12.39 List 4, length 5, 0.1 K 12.40/12.39 List 5, length 1, 0.0 K 12.40/12.39 List 6, length 1, 0.0 K 12.40/12.39 List 7, length 4, 0.1 K 12.40/12.39 List 8, length 7865, 245.8 K 12.40/12.39 List 11, length 180, 7.7 K 12.40/12.39 List 12, length 100, 4.7 K 12.40/12.39 List 14, length 1, 0.1 K 12.40/12.39 List 15, length 2, 0.1 K 12.40/12.39 List 16, length 600, 37.5 K 12.40/12.39 List 19, length 2, 0.1 K 12.40/12.39 List 20, length 2, 0.2 K 12.40/12.39 List 26, length 320, 32.5 K 12.40/12.39 List 27, length 1, 0.1 K 12.40/12.39 List 49, length 2, 0.4 K 12.40/12.39 List 76, length 2, 0.6 K 12.40/12.39 List 98, length 2, 0.8 K 12.40/12.39 List 101, length 2, 0.8 K 12.40/12.39 List 202, length 2, 1.6 K 12.40/12.39 12.40/12.39 ============================== SELECTOR REPORT ======================= 12.40/12.39 Sos_deleted=0, Sos_displaced=0, Sos_size=0 12.40/12.39 SELECTOR PART PRIORITY ORDER SIZE SELECTED 12.40/12.39 I 2147483647 high age 0 0 12.40/12.39 H 1 high weight 0 0 12.40/12.39 A 1 low age 0 0 12.40/12.39 F 4 low weight 0 0 12.40/12.39 T 4 low weight 0 0 12.40/12.39 ============================== end of selector report ================ 12.40/12.39 12.40/12.39 ============================== end of statistics ===================== 12.40/12.39 12.40/12.39 Exiting with failure. 12.40/12.39 12.40/12.39 Process 18349 exit (max_megs) Mon Jul 3 10:16:58 2023 12.40/12.39 Prover9 interrupted 12.40/12.39 EOF